carat22 45 mins is the MCT (Minimum Connection Time) for DOH so there should be no problem but it may be a bit of a rush if QR10 is late arriving.
QR10 arrives from a 'Secure' airport (LHR) so unless selected for a random security check you will exit the aircraft directly into the departure concourse. Most versions of the IFE display departure gates for connecting flights prior to landing so it is just a case of getting to the departure gate. should you be late arriving then there will be staff on hand to get you through to the new departure gate.
Even with security checks it is possible to make the connection.
Baggage handling in DOH is excellent and your luggage will most likely be onboard the aircraft before you are!