I'd suggest this: pull the engine apart and see what you've got. You might find that some things are in remarkably good shape, others need attention. When I took apart my TD engine, it was clear that the upper end was in great shape, good compressions, the cylinders were perfect within a half mil, and so on. The bottom end was a mess though, and it needed a new camshaft. So, I did a stock repair, not really a rebuild. I think that your choice of how to approach things should depend on what you have to work with. Especially if you are on a budget.
Same story for other parts. You don't need to do a tear-down restoration, but you want things to be in good enough shape so you can enjoy the car.
Also, realize that there are a lot of options for parts, not just Moss and Victoria British. British Parts Northwest has lots of MGB parts at good prices and plenty of parts are available on eBay. You have to be careful of eBay, as people try to sell things for stupid prices, but with some care you should be able to exploit it well. There are also a lot of specialty places that can be very helpful.