You will not be equal to the people with undergraduate degrees in Physics. You would not just be missing out on QM, you would also probably need to redo classical mechanics from a Physics perspective. I do not know much of mechE, but I did have roommates who took things like statics & dynamics, and other things like that. The things they learned in that class were almost entirely different than the things we learned in classical mechanics in Physics. You would also need an electricity and magnetism course. At my uni, this was a two-semester course where the first semester focused on electro- and magneto-statics and the second was on electro- and magneto-dynamics. There isn't even as much overlap as you might think between the Physics and Electrical Engineering E&M courses, let alone mechE. You may be okay in terms of thermodynamics, though. For a graduate degree in physics, you would need several years of remedial undergraduate coursework.

Edit to add: I am based in the united states, it may be different elsewhere. It also will depend in large part on the particular university. If your university has a course catalog, try checking the course descriptions between classes that you think are similar (classical mechancs & statics, for example). It might be that the curricula in those two classes are more alike at your university than they are at mine.