It all depends on how many people are on your flight and how many other flights come in around the time yours does, and how many staff are on duty at immigration.

Last August (during the Tattoo and other festivals), I flew into EDI on a Delta 767 from ATL that was 99% full. There were initially 2 agents checking passports, but a third stepped in after a few minutes. Unlike Gatwick, I did not see separate queues for EU and non-EU passport holders.

From the time I got up from my seat at the gate until the time I cranked the rental car was less than 45 minutes.