You should find two sets of engine mountings on the chassis, one for the "A" series and the other for the "B" series engine, and the Marina 1800 SC, TC and MGB engine will fit in with little or no modification provided the chassis has the mountings. I do not recall seeing any Mk 2's without both sets so you should be OK. The Marina 1300 and 1800 gearboxes are the same, but the bellhousings are different, so get a matching set. I don't know whether the MGB gearbox will fit, but I expect so. The rear section of the Marina exhaust systems are the same, but you will need a new front section from manifold to front silencer. I don't know about the MGB exhaust system. You will find that the rear axle ratio of the 1300 is too high for the 1800 and you will have acceleration to die for but no cruising speed - motorways and fast A roads wiill be a nightmare, as will your fuel consumption, so get yourself a Marina 1800 diff or whole rear axle as well. If you find an overdrive unit you will have to make up new gearbox rear mounting and have the prop shaft shortened and balanced.
You will need to swap the speedo head as well so that you don't collect those nice points on your licence.
Cooling systems are the same, as are front suspension and brakes, so no changes needed for the extra weight/power.
Apart from that lot it is a straight swap! And don't forget to tell the DVLA and your insurance company. The 1800 lump is a fair bit heavier than the 1300 so the handling will change a bit.
If you do decide to hang on to the 1300 engine and rebuild it (not a bad idea in my view) there is a plentiful supply of tuning and rebuilding parts to make it just as quick as an 1800 'dog (bearing in mind the 50kgs weight bonus of the lighter engine) from the classic mini market. Have a look at my hintsntips for what I have done to mine.