just pull the engine already and rebuild it.

Now, the question to ask - after another week of soaking in something - does it turn over?
What are you planning to do with it? keep/drive it or sell it?
with twenty years of sitting will the clutch release? brakes work? brake servo?
When I had an engine much the same as you do I took mine apart to see what all needed to be changed.  Engine was hot tanked, bearings replaced, time for a new cam, pistons,new oil pump, head work (springs don't like being compressed for years and years), water pump didn't move either, etc.

I would pour a beer or two directly into the owner, and remove the head. Too rare a engine to chance total destruction.

I run a 67 Cooper S block in my street car. It has had a long and storied life. It was std. when I first bought it. I have personally run it at std., .020, .040, .060 as well as 73mm. In it's lifetime it has been line bored, squared, bored and honed with deck plate and machine for zero deck. During it's very active Solo 2 period it was turned past 9000 many short spurts. It currently is a slow turning highway machine for the last five years. Lately it gives a puff of smoke on start up and #1 plug is trying to foul. I will sleeve her back to std without a thought. I only have three Cooper S blocks left, one at .020, one at .040 and this 73 mm. I'll pull the head first and maybe valve guide wear is at the bottom of my issues. Good luck, if you change your mind let me know and I'll hook up the trailer and start heading that way. Steve (CTR)

49'er,

I read from the top and what I don't understand is why the motor turned over 7 years ago but not now?   You loaded it up with MMO and now it's locked up?  Doesn't make sense to me.  However, let's say it frees up from your thorough efforts.  You ok with that #2 bore with the rust ring?  I'm not.  Not even for a garage queen. 
But it's possible something else is going on here.  Following my depressing logic, I encourage you to go the hard way and yank the motor, tear it down and figure out what the hell happened.  What a wonderful thing it would be to only hone it, ring it and throw new gaskets and seals at it, assuming the 3 syncro gearbox was 100% clean-as-a-pin.  Done.  Vrooom.  Yet I doubt this is what you will be faced with.
Pistons at .060?  What are the chances the crank is still at zero/zero?  Uh, zero, pun intended  I've been in your shoes, btw.  It's painful and time consuming.  But if you go the hard way you'll end up knowing a lot about S's and secure in the knowledge that your motor is rock solid and good for years to come.  In conclusion, it's an S.  It's worth it.  Even if you have to sleeve the block.

If you don't want to suffer through this BS?  I don't blame you.  But you can do it and good luck with it!

acetone and marvel mystery oil isn't CRC freeze spray. Acetone is OK to clean gum out of your carburetor, not so much for rust in a cylinder. Years ago I had a car that had gotten water in the cylinders via a leaking head gasket and it had sat for years and after spraying the CRC freeze spray into the cylinders for a week it freed up and ran. As far as those that insist you can offset bore the S block, good luck. Call David Anton at APT and ask him what the chances of failure are. Flip a coin that's the odds. You don't see any real S block 1380's for sale do you? There's a reason for that. Been doing mini's for 50 years, Good Luck.

At this point, go ahead and shove some lettuce down in there along with the apple cider vinegar... maybe some shredded carrots and radishes, too...

Pour Red Apple Cider Vinegar into all the cylinders and fill them up. Let it sit for a week then try rotating the engine by hand. Everything should be free to move.

first 49R where are you located

As a 1293 is +0.020 over that would make
       1310     +0.040
and somewhere in there is the offset boring moving some 0.010" from the center (between 2 and 3) and 0.015" towards the pushrod holes - verify this with Vizards - How to modify your Mini.

the largest sensible motor IMO is 1430 (a stroked 1380).  But then I had several of the Austin America blocks at the time to play with and doing an offset grind on the crank to get an 84mm stroke and 1.625 rod size was around $175.00 (1.75" rods non 'S).  Most advise against using the 'S block for big bore motors.  'S blocks aren't cheap but they can be found. I know of two or three available - but expect to pay around $1,000.00.

I am in Houston, TX. I have been using a 50/50 mix of acetone and auto tranny fluid as a penetrating oil. So far I have poured three 16oz bottles of this concoction into the bores. I am assuming that the number 2 piston is the problem because the penetrating oil is pooling on top of the piston but not on the other three. 
I have the driver side front wheel off the ground and in third gear. I check every few days to see if it will turn. The amount of travel is just enough to take the slack on the drive train which is not much. After that no movement whatsoever.

IF number 2 is at the top of bore so is number 3 as they run together as do 1 and 4. Usually it's the cylinders where the pistons are down in the bore a ways that tend to rust up. Get CRC Freeze spray and keep spraying it into all cyls. for a few days. The Freeze spray has ammonia in it which actually dissolves any surface rust or gum which may be causing piston to stick. If it's an "S" block and .060 over already you are at it's limit. If anyone here read Vizards book you'd know offset boring on an "S' block has a 50% chance of blowing thru into the water jacket and winding up with scrap metal. Unless you can find some NOS sleeves, then you can sleeve it back to stock. Some of the factory blocks actually had sleeves in them to fix casting issues.

I'd go easy with the wood block on the piston. I've cracked a piston trying that.  Keep loading it up with kerosene/ marvel oil or diesel fuel, etc.

Rock it in 4th gear to free it.  I'm with an S now that is stuck since storage with a bad hydro bag hose. Engine was running when parked, now locked up.

spank  great advise....   fortee9er  where are you on planet earth?? might be worth shipping to Steve Gibbs at Cooper Tune racing.. if you need engine rebuild..  any local machine shop can do the head work. costs usually about $250/350 and figure $350-500 for labor to replace the head gasket and head.. later bc

Thank you so much for your advice. 
I bought the car 7 years ago with a seized engine. Once I got it home I put Marvel Mystery Oil in the bores and after a week I was able to roll it in gear. I put more MMO in the cylinder bore and let it sit for a number of years. I have way too many projects and finally got back to the Mini recently. The last tag/inspection is dated 2000. So the car has sat idle for 19 years.

Step 1, pull the goshdarn head already. It's the price of some gaskets versus the price of a numbers matching cooper s. Even if you did free it up with it all closed up, you really dont' want to have rust paste sliding up and down the cylinder walls. Removing the head will allow you to wipe all of that out before it is rubbed against the walls at thousands of revs per minute till they burn of or shoot out the exhaust .

Step 2, remove as much rust and crusting as you can with handtools, move to green scrubby pads, and teflon/plastic pokey thingies and even razor blades between the piston and piston wall if you feel personally comfortable with that.

You can also use a wooden dowel and a hammer alternating wacking on the top of the piston from distributer side to exhaust side to rock piston a few thou that is the skirt clearance. This will both help to unstick the rings from the bore, and unstick the rings from the ring lands of the piston.

at some point you'll be able to rock it just with your fingers a little bit.

Once you can get the piston to move down the bore a little, assess the situation / bore condition. You may be able to live with it after you clean it up with a cup brush or a green scrubby or even a "quick and dirty" light hone while the piston is still in the bottom of the bore and engine in the engine bay.

There's doing it RIGHT and there's just doing it. But both methods require the same beginning-- pull the head.

How long was the car sitting without having run?  

While I would be mentally prepared for a rebuild you might still get lucky.  You have very little to lose by gently rocking the car forward and backwards in 4th gear.  

If you do find bore damage when the piston frees up, you can get liners for the block.  They are not cheap plus you have to have them machined after installation.  However, as Kermy said, if you want to keep the original block the liners may be your best option.

A 60 over block has a capacity of 1311cc. The largest sensible motor is a 1380 but I believe the block has to be offset bored to achieve this which I think is too late now that its at +60. If your current block is a Cooper S block it will be very difficult to find another one.

I was trying to revive my Mini Cooper (1275cc)  and after chasing and fixing some electrical issues I found that the engine is seized.
After a week or two of pouring penetration oil into the spark plug holes the engine would not rotate. I used a boroscope to look into the bore and see if I could find the problem. The problem looks to be the number 2 cylinder, which is at the top of the stroke. Another tell tale was that while the penetrating oil had traveled passed the rings on all the other cylinders the fluid had pooled in the number 2 cylinder.
I have come to the conclusion that it is going to require a full engine tear down and rebuild to fix this problem. However if any of you reading this have any suggestions please let me know as I would greatly appreciate it. 
While I was looking into the cylinder bores I could read the top of one piston. It was an AE 60 over piston, obviously not a virgin.
If the number 2 cylinder bore is damaged what is the maximum safe over bore for a 1275 cooper block?
Should I look for a good block or have this one sleeved since this is a numbers matching car with a Heritage Certificate.
Thanks in advance for your advice.