hi, im fairly familiar with Austins, but not particularly Healeys, but, i would suggest it is a bad coil (can be a real bitch to remove so prepare yourself)or, If your engine has been rebuilt at one point, maybe check the timing chain has been put back correctly, mine wasnt!! One last thing, try extremes of temperature on the carbeurettor. DO THIS AS A LAST RESORT ONLY- CAN BE DANGEROUS. For some reason, on both my Austin 10's and on the austin 7 if it was left for a bit, it would not start. So, someone recommended this technique, and it worked first time! Im not sure of the science behind it, but it worked anyhow!!
1) turn engine over a few times to fill carb with gas
2) prime the fuel pump (if accesible on a healey??) to flood carb with fuel
3) Use something VERY hot rested on the carb bowl. I used hot cooking oil, as this was suggested to me. Just literally hold the pan under the carb
4) use the other extreme. I used an ice pack for about one minute
5)turn her over, VROOOOOMMMM
BTW, this worked for me, but it may be worth a try.
You must think english are crazy by now...