I had the same problems when I tried to join 7 years ago and was told that I couldn't be a WE officer with a degree in physics as it was a science degree and not an engineering degree. Surprise surprise, i have now been in for 6 1/2 years with 6 of them as a WE officer - i managed to transfer when i was at BRNC after getting in via the FAA route. I don't believe that the recruiting service can turn you away without an accredited degree, because what are they specifying that you be accredited for? My degree it not accredited with the IET but I am still a member and have the option to become a Chartered Engineer if I attain an MSc in Engineering (which is possible with the Navy). The Navy does not stipulate that we be Chartered Engineers therefore i don't see how they can stipulate that you must have an accredited degree.
Further to that, if you have a grounding in physics and have done any nuvlear phyiscs, i'm suprised they didn't tryo to get you to sign up as a submariner, as this involves completing the Nuclear General Course which you should hopefully be sypathetic towards completing.
Don't write yourself off as a warfare officer, you'll get bored cos driving ships is about getting up in the middle of the night and knowing how to draw triangles, it doesn't get interesting until you become a principal warfare officer. if you want to do engineering, you could go the WE route as you can become a PWO(WE) for a spell if you want.
Hope this helps.