Engine Transplants
Down here in New Zealand, the most common engines fitted to Minors are the Morris Marina 1275 and the Datsun 1200 engines, although, the much more modern and powerful Toyota 4A-GE Twin Cam 1600cc engine is making an impression on the Modifying scene.
The engine fits surprisingly well into the Minor engine bay. It's almost exactly the same length as the Datsun 1200, maybe even a little shorter (the ancillaries like water pump and fan blades stick out less).
The 4A-GE puts out around 110bhp in standard form, so if you follow this path you should expect to have to change the rear axle, and obviously the brakes all 'round. The poor old Minor diff just wont take the power and the front drums are hopelessly inadequate for slowing a car rapidly down from a possible top speed of around 130mph/200kph !
The choice of differential is up to you, but the humble Vauxhall Viva does an excellent job of coping with the standard power output of a 4A-GE. If you are looking for more power, then think about buying an entire RWD donor Corolla. Buying a donor car will ensure that you get all the right bits, the diff will be designed for the engine, and with a bit of prudent shopping around, you may even get a disc braked rear end. For these try finding a Trueno or Sprinter (model names may differ outside of NZ).
If buying the engine & box by itself, make sure that you also get ALL of the plumbing. This includes all wiring from the engine back to the plug fittings, the electronic ignition, fuel injection unit and the 'Black Box'. The Box is VERY important, 'cause it's this that controls all the electronics, ignition timing, knock sensor etc etc. Buying a Box by itself later could prove expensive.
Fitting of the engine, gearbox and new diff should probably be performed by a professional. Fitting a new diff involves manufacturing new spring & axle mounts, and the WOF/MOT men will look closely at the work done. My advise when it comes to welding important bits - Let Someone Else Do It!
Plenty you say? Want more? SUPERCHARGE IT!
The 4A-GE is also available in Supercharged form, as in the 4A-GZE, as fitted to the MR2. Fit one of these babies into a Minor and you'll have one awesome piece of machinery! I have yet to see one in-the-steel, but I'm sure one can't be far away.
The 4A-GZE puts out around 150-160bhp. These engines also perform well if converted to use 'Old-Style' carburetors. An Off-Roading friend of mine removed ALL of the electronics and fuel injection, made up a casting to accept a Holley 350 carburetor, fitted a smaller supercharger pulley and locked it in place so that it operates at all engine revs giving about 15 pounds boost at all times! Driving through a VW gearbox (the engine sits in the back of the Off-Roader) it dynoed at 160bhp AT THE WHEELS!
Fit a 'GZE to a standard-looking Minor and you have an awesome Sleeper!

The Japanese, with their know-how of taking a good idea and making it great, transformed the ageing BMC lump into a fairly good performing, high(ish)-revving engine of 1170cc. This powered a range of Datsuns from the early 1200 series of cars through to the 120Y and Sunny family-cars.
When transplanted into the Minor, the slightly lighter body and the lower ratio differential results in a liveliness from the Datsun 1200/120Y engine that hadn't previously been seen in it's mid-size vehicle incarnation.
A Datsun-powered Morrie is a Morrie transformed; capable of cruising happily at 115kph/70mph all day, and even 170kph/105mph with little modification.
When fitting the engine and gearbox, you may also find (depending on how high the engine is placed in the engine bay) that the top of the gearbox fouls on the Minor steering rack. This is easily fixed by shaving some of the excess aluminium from the ridge that runs down the top of the box'. In extreme cases you may have to actually cut a hole in the top of the 'box. Don't worry, the 'box is not load-bearing and wont collapse (not unless it's a really HUGE hole, in which case there's something wrong with your setup elsewhere!!). If you do end up opening a hole in the box, you may like to think about cutting out a piece of sheet aluminium and riveting or glueing it over the hole to keep dirt and grime away from the clutch mechanism.
When positioning the engine in the engine bay, it will be necessary to push the engine as far back in the engine bay as possible in order for the fan blades to clear the radiator.
DO NOT DRILL HOLES OR WELD THE ENGINE MOUNTS IN PLACE YET !!
Mark out where you think the engine mounts should be positioned, but this may have to be shifted slightly when fitting the gearbox cross member and the drive shaft.
The later model Datsun 120Y engines had the Cross member mounting positions located further back on the gearbox than the earlier 1200-series engines. These later gearbox mountings are in almost exactly the right spot for bolting to the existing Minor Cross member. Just a couple of extra long bolts and a rubber bush is required to secure the gearbox in place.
At this stage I recommend you offer up the Datsun drive shaft to the Minor diff'. It should bolt up directly to the Minor diff', although you may require slightly longer bolts.
Fit the nose of the drive shaft into the back of the gearbox and adjust the position of the engine & 'box so that there is 15 - 20mm of slide movement on the drive shaft splines. This will then give you the required position of the engine mounts. Of course this may have positioned the engine too far forward and the fan blades will foul the radiator. In this case you will either have to fit an electric fan (I suggest in front of the radiator, behind the grill slats), or find yourself a thinner core radiator from a late model vehicle (the standard Minor radiator does an excellent job of cooling the water, and a thinner one shouldn't provide any significant cooling problems. If you're worried, you may like to fit an extra-wide radiator).
Well, that's the basics of it. I know there's a few murky bits in the text, but because of the infinite number of combinations of engine, gearbox, drive shaft, engine mounts and Minor models, it's impossible to know exactly what you require.
In general, you can expect to have to fit and remove the engine and 'box at least half a dozen times before you get everything in the right place.
Maybe it's easier to get someone else to fit it ... ?