Since your solenoid appears to activate your electricals are working (which is the most common cause of O/D problems). Minimum oil should be enough to prevent damage, so my first guess is your oil got aerated, which could prevent sufficient operating pressure (like spongy pedal when brake fluid gets aerated). Let the car sit for a day or so, then try again; if it still doesn't work then it could be the hydraulics, but those don't usually fail without warning (slower and slower engagement).
Another (better) possibility is your solenoid is working, but the lever that rotates the cross rod that opens the activating valve is slipping on the cross rod; this could happen suddenly. You didn't mention what model Healey you have; earlier cars have this lever exposed, but later cars have this enclosed, and I don't know if you can get the cover off the box in situ. If the lever moves but the cross rod doesn't you have your answer; you will need to adjust and tighten the clamp on the lever.
I'm not sure what you mean by "Car has very low miles;" some have taken to resetting the odometer after doing a restoration (it's not hard to do). Healeys with less than 50K 'true' miles are pretty rare.