Hi DAR,
I think it is two separate problems.
First is fitting a later model 5 stud hypoid type axle. This is mainly a matter of getting & substituting all of the right parts into your car. It should be fairly straight forward. Easiest would be to first compare parts lists for the two cars & find out what details are different. Brackets, bolts, clamps, brake assemblies & drums, & misc. bits. I have not done the swap. Hopefully, someone here who has done it can advise.
Second is fitting a new ring & pinion to the new axle center section. This could be done at home but I would advise you to have a good shop do it. If they are not experienced with the Healey axle, supply a good shop manual to the shop. You might be fortunate enough to find a shop that has Healey experience. Mike Lempert who supplies 3.54 ring & pinion sets might be able to advise of a shop to send the center section to for the work. Others might be able to advise on a good local shop.
There are various shims & spacers which have to be assembled exactly right to get the gear mesh & the clearances correct. As mentioned before, you "might" be lucky enough to find a complete axle assembly with the correct axle ratio, but it is a long shot.
If you don't have one, a copy of the AH 100 Service Manual with the BN2 supplement will be very useful. It is available as a reprint from various sources.
D