If your Healey uses the push-button solenoid shown above, and if it's an aftermarket or late model part... I am painfully aware of problems with that item.
The push-button solenoid was used on a number of LBCs in the 1960s. The current batch of parts has a design problem. There is an "E"-ring on the solenoid plunger shaft that is supposed to push the copper contact into position to pass starter motor power. Over time the E-ring loosens and falls off the shaft which prevents the contact from moving and thus... no starter power. I've also had one of these where the contacts spot welded themselves together which forced me to pull the battery cable to stop the engine from cranking.
Of the three I've purchased over the past 5 years, none has lasted more than about a 18 months. As I mentioned, I've seen them both fail to engage and fail to disengage. It's quite unlikely that you wired the solenoid wrong (as long as you didn't put both big cables on the same lug). It is quite likely you've got a problem with the solenoid itself.