Nope...had to replace the backplate.  Even after I got the old backplate off the car I could not get it unstuck.  Oh well.  Just replaced it yesterday.  It was an easy enough job.  So my Mini is back on the road, and with some new Cooper mirrors too   I figured why not, if I gotta order one part, why not a few.

Thanks everyone for the help!

Surfin, did you ever get it unstuck?  Just curious what the current status was.

If you have the equipment, this has worked for me. Start by wire brushing all of the adjuster threads you can reach and apply penetrant. Get an 8-point socket that fits the adjuster, Sears usually has them, they are made for square nuts. You can reach the adjuster through the subframe with enough extensions. Use an air impact wrench set at low torque / low air pressure and let it bang on the adjuster for 5 or 10 minutes. Turn up the pressure and do it again add penetrant occasionally. If your impact wrench doesn't have a torque / pressure adjustment you can buy an  inline adjuster. Eventually it will start moving, when it does reverse directions a few times.

Kelley

Thanks for all the input so far.  I've been working it at for awhile now.  By the way, it's the right rear drum that I'm having the problem with...sorry I forgot to mention that.  So far, I've tried using PB and a lot of heat, but to no avail.  Since even if had to order a new backplate, it won't get here for awhile due to the long weekend, I'm going to try taking it off and working on it that way.  Since I have to pull apart the hub to get it off, are there any parts I should probably replace?  Is it likely that the bearings will need replacing?  And, in the event I need to replace the backplate, does anyone have one lying around?   Thanks again for all the help!

If you can stand to do without it for a few more days, possibly a week, then here is what I would do.  Take everything off the backplate and find a container that is large enough for the backplate to sit in.  (like a Tupperware container)  Anyway, go to the store and buy a gallon of Apple Cider Vinegar--the yellow stuff and it is very cheap.  Put the backing plate in the container and put enough of the vinegar in the container to completely cover the backing plate.  After a few days the adjuster screw will turn out very easily.  No heat required.  I had an engine that was stuck solid from water sitting in the cylinders.  Tried everyway possible to break it free with no luck.  Put the best Craftsman breaker bar on it with a 6' pipe and still could not turn the engine over.  Tried PB blaster, etc, with no luck.  Had to get the engine to turn so we could remove the flywheel properly.  Anyway, poured the vinegar in it and left it for a week.  Came back and was able to turn the harmonic balancer by hand.  Easy peasy!  Apple Cider Vinegar eats the stink out of rusty stuff.

You never mentioned front or rear. You also did not reply to my post about front end parts and brakes.

For rears I remove the backing plate from car then strip all the brake parts. I believe the last one I dropped a nut though the hole on the shoe side and welded it to the cone down though the center of the nut. While still hot I put the impact to it and backed it right out. It cones out the shoe side of the backing plate. Run a tap though and install new adjuster. Remember to lube well. Failing that I'm sure I have a backing plate we'll need to know which side.

Steve

Many years ago I had the sam problem, (rear brake adjuster). If memory serves me correctly I drilled and tapped a hole through the complete adjuster (5/16" (LEFT HAND THREAD)) and inserted a left hand hex bolt through the back and went at it. Eventually I won & it came out.

A lot of of work but I had time on my hands for once.

Martin.

heat it up with a torch then spray it with a cannned air ONLY SRRAY

THE CAN UPSIDE DOWN, THE LIQUID GAS IN THERE COMES OUT

COLD AS  ICE. The expansion and contraction may loosen it for you.

I couldn't find it on the site, but don't "they" still make the repair kit where you cut off/drill out the old adjuster and attach the new bits?  I did this years ago and it was easy as pie.

I heard a trick from an old mechanic (happens to be my father in law) he says for those really tricky ones, to heat the part, in your case the adjuster, then shave a candle down to where there is a point that will touch the threads.  Just touch the candle to the hot part.  Then heat again, touch again, etc.  He says the heat will really suck the candle wax down the threads and allow it to free up.  I've not had the opportunity to try it (yet), but I watched him recently do it again on a stuck bleeder valve on an old VW drum brake.

Willie_B, do you mind posting that picture in higher resolution or if you can email a copy to me.

Thanks.

--------------------

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had that problem with my mg,  wound up using pb blaster,   love that stuff.  pb and tapped with a wood mallet.  eventually it broke free enough to wrench out.

The last one i had i took the backplate off the car and put the adjuster in a vice and worked it with heat and penetrating oil until it broke loose, i was getting nowhere with it on the car and i would have ordered a new one but needed the car running the next day.

However you effect the repair, try and find a can of this:

//www.castlepackspower.com/index.cfm?Page=Castle%20Brake%20Life

Comes out of the can like cheese whiz.  
Coat the threads and work it in 'n out a few times. Seems to hang on forever. Even in a wet enviroment. Also makes a good dielectric grease. A genuine multi tasker.

Can you weld a nut onto it?...

Front or rear, left or right? It makes a difference when replacing. Peal back the rubber boot on the wheel cylinders and see if there any fluid under it. If so, the cylinder is leaking. Take good photos before removal as it matters how the springs and shoes go on.

Simple brake maintenance gone sour   2 weeks now with my mini on stands because of miscellaneous parts braking...Though I'm sure this one is a common problem.  My brake adjuster screw is seized.  I've tried spraying it, torching it, and nothing.  Won't budge.  Any other suggestions? Worse case, do I have to replace the whole backplate?  If so, anybody have experience as to what other parts will likely need replacing while I am at it?  And...does anybody have a backplate they'd like to get rid of?  I want to get my mini back on the road!

Thanks everyone!