Ironskull,
It would help some more if you elaborate on your research interests. My suggestion, forget about mechie phd, continue with physics.
As previously said, you will have to take undergrad mechie classes and trust me if the graduate chair or advisor is a bit stringent you will invest two years just to catch up with those classes. There is no hard and fast rule on how many undergrad classes you should take, I have seen ugrads from applied maths doing engineering masters without taking any undergrad classes. This is something you should discuss with graduate coordinator/ chair or advisor before applying.
The easiest way for you to be sold in engineering would be to target profs who are doing more theoretical and experimental research in mechanical engineering. Things like algorithm development for computational fluid dynamics, unsteady aerodynamics and heat transfers should be more easier for you to adapt to. I can't say for sure but such profs should love to have physics ppl. If you go into the more applied aspect of engineering, you might hate us engineers for the amount of abstraction we use in development.
Another route is to go for experimental physics rather than engineering. You will be more into theory than engineering and also have some hands on experience. For the aforementioned topics there is minor differences between what engineers do and what experimental physicist do.
Now, the most important thing is to know exactly why you want to do a Phd. If you are into Phd. because you want to teach as a mechie faculty, physics phd might not help. But if you are into Phd. to work in the industry as a researcher, it might not make much of a difference what topic you get your Phd. in. Essentially, Phd. is training in research. It prepares you to conduct original research without supervision, meaning even with a physics phd. you have enough knowledge to dig into engineering theories and propose improvements (citation needed). If you look into job openings at places like Boeing, Lockheed and Raytheon, their requirements say something like "Phd. in Engineering, Applied Maths or Physics" ( I might be wrong, go and check for such requirements in your field). I would be really pissed if I got my phd. in mechanical engineering after doing two years of additional ugrad course, only to realize that the guy working with me has a phd. in physics.
Damn, Its getting long. To cut to the chase, do some more research, some intense soul searching, talk to some profs and let us know what happens.
Good Luck.