Upvote:4
I'm going to assume the bike is not a folding bike, otherwise you would have specified it as such. Given that, you can't catch London buses or the Underground from Heathrow as the former does not take non-folding bikes and the latter not from that station. Here are your options:
I didn't bother checking private buses because IMO the train transport would trump it, given the shorter changes required (National Express, for example, drops you off near Victoria Station instead of near Baker Street).