An oil leak from the diff nose will cause the rubber of the brake hose to swell and fail MOT. The EP oil (extreme pressure additive) will rot the rubber. Seals are about $5, but to under the nut under the universal joint is hard. It is VERY tight but one undone the deal is easy to change. Re-torque that nut up to the right amount. (I put my foot on the spanner, my hands around the spring, and heave as tight as possible...) Top up the oil.
Back Axle Whine... Well they all whine. Some more than others though. If it whines loudly, check the oil level. EP90 is ideal oil. Axles get very very hot as normal and do not panic if you find that you could fry an egg on yours after a long journey. If the whine is very loud, perhaps fitting another secondhand unit will be better. Use the right ratio, see your manual.
Oil Leak on the BACK of the brake back plate is a worn out half shaft bearing seal. Signs are that the bearing is shot as well. Kits are available to cure this.
Oil inside the brake drum comes from the seal between the halfshaft and its bearing. Early cars only had a gasket. Oil on the brakes require the fitment of new shoes. No attempts at cleaning will make them safe again.
The large nut on the ends of the wheel bearings are 1 and 61/64ths. No common wrench will work. You must order this from an Austin Healey Specialist like Moss Motors.