Unless your flight is late you shouldn’t have to make a run for it. I haven’t flown into DOH from EDI, but if you come in from an airport considered safe, they let you go directly into the departure area instead of making you go up a floor to go through transfer security screening and then down again. Saves a lot of walking and at least 20-25 minutes at the best of times. With a 50 minutes connection you might even have time to nip into the Al Mourjan Lounge if your onward gate is not at the far end of a pier. They have electronic screens in every pier which allow you to scan your boarding card and then show you the route to your departure gate. Just use one once you have disembarked and you’ll know whether you have enough time to go to the lounge. If you do, ask a lounge dragon on your way out to phone and check whether your departing flight has actually started boarding, this will save you from leaving the lounge early and unnecessarily waiting at the gate.
Added after reading SGJNI1961’s post:
If you ever come across a crowd at the transfer screening point, don’t let the uniforms force you into the economy que which will be going in towards the right or the left depending on which side you are coming from. Go further on towards the middle where you will find the priority lanes hidden away! Most of the time you have to ask someone to open the cordon by showing your premium boarding card holder given at check in. This is also where they take the wheelchair passengers in from if you otherwise miss them.