When on 1 ticket, the connection time is fine. If your incoming flight is more or less on time, you will easily make it to the gate of the BA flight in time. I would say that your incoming flight from DOH needs to be 45min late to jeopardize you making it to the BA flight.

You can use the dedicated short connection transfer/security checks (hereby avoiding the often long walk to the main transfer security checks). Look out for the signs when you start walking in the terminal after disembarking the aircraft.