If it is on the same PNR, IATA rules require them to interline. If it is on two different PNRs, then it is up to the first carrier (ie - United) as to whether or not they will do it. BA will only do it, for example, if it is another OW carrier.

You will not need to pick up your bags in Heathrow. They will be tagged international, and you will need to go through the appropriate customs queue at GLA.

Quickest way from T1 to T3 is to walk underground.