On the spiral bevel BN1 axle, this axle is not like the later & more familiar hypoid designs:
The ring to pinion backlash is set by shims which set the side to side location of the carrier/ring gear assembly. The pinion depth in the housing is fixed & pinion bearing preload is set by shims. The pumpkin MUST be removed to properly set up the gear mesh adjusting shims. On the later hypoid type axles (late BN1 & up) these same pinion shims are used to set a specific depth of the pinion in the housing.
On rare occasions where things have been operating normally & suddenly get out of whack, the pinion preload setting shims have been known to break up & allow the pinion to go out of depth adjustment. This is the only instance where the assembly might be corrected for excessive backlash without removing the pumpkin. I certainly doubt if this is your problem, considering that it was just worked on.
Most likely - someone just set it up wrong the last time it was worked on. Either pinion preload or ring gear lateral position. I would return it to the mechanic & insist that it be reassembled correctly & with the PROPER .005" to .008" backlash between ring & pinion. Obviously, ring gear runout must be checked for a maximum of .002" with this small backlash. The correct shims for this axle are not so readily available, but a good shop should be able to come up with something. Don't accept any attempts to "fix" it without removing the pumpkin from the axle housing, or the story that it will get better with a few miles on it. It won't. As I said before, if you run it very long in this condition, it will be permanently screwed up.
The front pinion seal is a separate problem. Sometimes the seal wears a slight groove in the pinion flange & the new seal needs to be set at a slightly different depth so that it can seat on a "fresh" pinion flange surface. Oil can get out between the flange inner splines & the shaft splines. It is a good idea to apply sealer to the splines when assembling.
If I "have" to leave the car with a shop that is not familiar with the car, I leave the factory shop manual with them. Many shops do not have the needed information available for the particular car & have to guess at or assume things. If you don't have the manual, get one. The reprint of the Austin-Healey factory service manual, BN1 with BN2 supplement is very good. British Car Specialists in Stockton, CA has it along with a pretty good selection of parts.
Sorry you are having this problem.
D