LHR is a problem as they won't let you transfer to another terminal if not flying from there. The only option is to exit T5 but even then it wouldn't be worth it - getting back through immigration/security is a royal pain.
Your best bet is to stay at T5 (it is very nice with excellent facilities) and spend the money on toys and food.
Perhaps some people on the BA board would be willing to guest you in (unlikely with an infant though).