"me becoming employed at a AAA studio, and actually needing c++, is really unlikely to happen"
I dont work in the industry (Still a student), but I would have to say I disagree with the above statement.
I have no idea what your physics degree consisted of, but I guess you have a solid understanding of motion, forces, vecotrs, etc. In addition, you may have all sorts of maths knowledge which applies to 3D games / graphics / physics engines. At the very least, you will easily be able to learn the 3D math stuff - and at a *MUCH* better level than a student. Maths is a key skill in the games industry without a doubt. Mathemetics For 3D Games Programming and Computer Graphics by Eric Lengyel is a good place to start looking/comparing your skill sets.
Personally (as I did in formal education) I would start with C (Being able to make some console apps/games) and then learn C++. C and C++ are similar and most books dont go in to classes from chapter one (Unlike Objective C, for example, which is a class heavy language).
From their, the world is your oyster. You can apply your degree and aim to develop some physics systems. Can learn Direct3D/Open GL and go down the graphics route. Learn sound programming, networking, etc.
No Games Programming degree/Computer Science degree can teach everything. Mine teaches C++ and thats literally about it (and we have a decent employment rate within the first 6 months). The best games programming degree in the UK (Teesside if im not mistaken) hasnt tought any content that I havnt been able to do in 4/5 months of studying Direct3D11 by myself (And I would say that learning something yourself is more important than the degree - indeed, everyone I have spoke too says the first question in an interview centers around things you do outside of education. Plus, imo, teaching something yourself gives you a much more rounded knowledge whenever you finally get it. Uni has wayyyyy too much hand holding to be good for you).
My mate did Computer Science (Newcastle) and, well... I wasn't impressed tbh. Defenatly not for games programming anyway.
So yeah, why not? If your looking for a career change, then your not in a terrible position. Most junior jobs state that they are looking for people with "Computer Science (and games programming) or maths related degree" - You do have a maths related degree.
Finally, learning c++ and games related programming can take time to do correctly. C++ is a hard language. Its of course worth noting that learning C++ as a second language is MUCH easier than it being your first. C#/Java (I did objective C. But I'm a mini mac fanboy ) I have heard is a good start if you want another langauge first .