Whether or not immigration is putting a strain on it, I can tell you that it is much worse.
And I can tell you that the effect of immigration is minimal and, in my opinion, as a regular M25 traveller, it's not as bad as it was in the early days. It's still pretty bad, but frequently the traffic is moving even at the worst of times.
I drove all the way around it in one go on at least two occasions between about 2000 and 2010, and did numerous other journeys.
I've been driving round it since the late 80's and I've encountered stationary traffic consistently since then.