Studying for an MEng is notoriously tough. It's a level of study that seperates those who want a solid foundation in Engineering (BEng) to those who have decided to study Engineering to a level that extends beyond undergraduate technical knowledge and builds on further professional skills required for industry. From my observation, Queen Mary normally expect at least an A (if you want study the MEng course) in a technical/science subject and good grades from other relevant subjects. Although, this does depend on the intake for that year and the level of demand like any university. Nevertheless, if you get good grades from your first or/and second years at BEng, then at your request, with the Head of Departments approval, you can switch over to the MEng course.
Personally I think a lack of Physics, is likely to make Mechnical Engineering particularly difficult, especially as the course is focused on the application of physical sciences to the real world. Not having studied Physics is also likely to lessen your chances of getting a place. Although In some other types of Engineering and some particular university courses, it's possible to get away with not doing Physics, since really if your Mathematics is sufficient you'll learn the principles while on your degree. Otherwise Maths is really the key pre-requisiite. Doing Physics will just help alot.
You could gamble and apply to the BEng course and if there are places left, which in Engineering departments there usually is, then you might still get a place. Otherwise, if your passionate about Mechanical Engineering, then someone somewhere will definitely let you in.
As for the Mechnical course, I couldn't give you a first hand experience, but hopefully QMUL's Engineering pages will help: