Hi and Welcome,
It sounds as though you have the famed Scuttle Shake most of us have experienced and some are still experiencing.
First, does your tremor start somewhere around 50 MPH or so and seem to disappear somewhere around 65 MPH or so?
Does your shake seem to change in intensity from time to time (sometimes severe and sometimes very light)?
These are some of the signs of Scuttle Shake which is often related to imbalanced brake drums. Although this tremor could be caused by a number of balance issues, the most common and overlooked is that of rear brake drum balancing. Factory and most replacements drums are not pre-balance and the Healey's rear brake drums are rather large and heavy. Since their size and weight is non-trivial, this imbalanced component transmits its issues through the springs and drive train to the car and shows up around 50 to 65 MPH. The severity of the imbalance, and resulting shake, is not constant as the left and right drum will constantly shift their relative position and counterbalancing relationship via the operation of the differential.
Since drums of this size and weight are no longer commonplace, finding a servicer to correct and balance these drums is not as easy today as years back when drums were commonly used. Look in your location for a servicer who can perform drum balancing as it requires securing weights to the drum in a secure manor that allows for heat expansion and contraction. If no one can be found, many Healey owners have gone to Hendrix Wire Wheel (hendrixwirewheel.com/drums.html) for this service.
Be aware that many local services will indicate they can cut the drums and this should solve the issue. However, cutting usually does little to resolve the problem and you need someone that is experienced in applying balancing weights to the drum.
Hope this helps,
Ray(64 BJ8 P1)