Travelling from PHX to GLA next week.

Ticket 1: United PHX-IAD-LHR
Ticket 2: British LHR-GLA

United told me over the phone I can check bags all the way to GLA, but I won't be convinced until I see the bag tickets, what with all the stories here about agents making their own rules.

Can anyone bless me with their experience or wisdom on this proposition?
I've seen quite a few threads where BA refused to interline bags AFTER the BA flight; what about receiving bags FROM UA?

Also, in the event of having to collect bags at LHR, is a 3 hour connection enough time to deplane at T3, passport control, reclaim bags, customs, hussle over to T1, check-in, security, blah-blah-blah..?

What's the quickest way to get from T3 to T1?

Many, many thanks.