Also have the mechanic inspect and adjust the fuel pump's points (assuming original SU is in the car). Have him put in new points--or at least file and clean the old ones--if there's any doubt. If the mechanic doesn't know how to do this find one who does. Or, order a new fuel pump--I recommend the SU solid state--and have the old one rebuilt for a spare. The fuel pump is by far the most likely thing to let you down; you can make it home if the O/D fails, for instance, albeit slower.
What kind of shape are the tires and wheels in? Have the mechanic inspect the splines on the hubs (wheel and axle). If they're seriously worn you'll at least need to drive very gingerly to avoid serious problems or, better yet, arrange to get them replaced before the trip (you'll have to do it right away anyway). If you replace the rear hubs have him pull the drums and inspect for oil leaks and brake shoe condition (not a bad idea to do this anyway, and not terribly difficult). Have him check the rear wheel bearings (if they're bad they won't turn smoothly).
What about the tires? If they're more than 10 years old--AT MOST--get new ones (again, you'll have to do this right away anyway).
Have the mechanic tighten all shock bolts (35 ft-lbs is good). Have him tighten all fuel line fittings (at the carbs and at the fuel pump). Check the U-joints.
If you do have him repack the front wheel bearings have him inspect the stub axles for cracks.
Change the coolant if it hasn't been done in few years. If a serious amount of crud comes out overheating--depending on ambient temperature--may be an issue.
Make sure all the gauges--fuel, oil pressure, coolant temp, speedo and tach (you can live without it)--work.
How old is the battery? If past its warranty age I'd replace it.
The Austin-Healey Club USA publishes a membership 'Resource Book' which includes names and contact info of Healey owners who are willing to help out fellow Healeyites. You might be able to get one:
https://www.healey.org
I take a 3-4K mile road trip every year in my BJ8, and I've never been stranded although we have had some problems (see: fuel pump). Since I know my car's history and condition I don't do all these inspections every year, but they all get done at one time or another. Reputation aside, these can can be very reliable cars, else I wouldn't drive mine 100 miles from nowhere in the Nevada desert.