"What is the minimum connection time between BA (Terminal 1) and AA (T3) at LHR."

75 minutes


"I have a flight arriving from Geneva 8.45am and a flight to Chicago at 10.05am. I am a UK citizen so am comfortable that I can easily make it, but my colleagues are US citizens, in which case I assume its quicker for them to take the airside bus."

It is probably quicker for all of you to take the airside bus unless there is an awful queue at the FCC. When you get off the bus, you will have to go to the AA check in desk in the FCC to get your boarding pass since AA in its wonderful spirit of cooperation around its OW partners will not let others issue boarding passes for their flights

"Second question - will BA in GVA book the checked luggage through to ORD? If not, we will have to collect the checked bags making it even harder to guarantee the connection."

Yes

"Finally, if I buy two separate tickets (AA - ORD-LHR return) and BA (LHR-GVA return), will BA check the luggage straight through."

They don't have to, but I have never had any issues with them doing through checking on separate tickets

Dave