I'm not a local, but I can tell you that last summer we
- Took the Heathrow Express to Paddington
- Paddington to Embankment (Bakerloo/Brown line)
- Walked to Covenant Gardens
There is of course an underground station for Convent Garden, but depending on your timing (and whether or not you're schelping bags and children), it may not be worth waiting for the transfer. Hope that helps.