These comments are making me cringe a bit due to the inaccuracies. I'll try to address them from the bigger matters down to the smaller ones.
Myth #1: Mechanical Engineering as a degree gives you a broader variety of options compared to Aerospace: No, and often it's the opposite. A good Aero degree will cover mechanical engineering sufficiently that it won't be an issue. The only things I missed out on compared to the mechanical engineering grads were FEA, railways, and tribology, the latter two were options for mechanical students, and I missed FEA because I specialised in control engineering instead of mechanical or materials. On the other hand, in an aero degree you can study a decent amount of Control and Electrical engineering which will give you options in those jobs. Aero also generally has a much bigger programming component so it's easier to get into those jobs which require programming skills.
Myth #2: BEng followed by MSc Aerospace Engineering is a drop in replacement for MEng Aerospace Engineering: No, you will miss out on the group project, industrial projects if they are offered by the university, the control/electrical/programming aspects, and if not these things then you will miss out somewhere else. You also don't usually get to do many 4th year MEng modules as they'll need to cover lots of 3rd year content or MSc specific modules which also cover earlier content. This often means you'll do e.g. Aero propulsion, but not advanced aero propulsion. Ironically the higher level modules are sometimes available in the Mech MEng.
A better option can be a more specialised MSc e.g. Aerodynamics. Accept that you'll lose out in some areas and instead specialise in one particular area to get on par or beyond what the aerospace MEng students studied.
Myth #3: BEng then MEng: Not unless you want to take 7 years to graduate. MEng is undergrad only, you want an MSc which is postgrad.
If you sign up for an MEng but want to do a BEng then you often have to let them know before the beginning of 3rd year as there can be some differences in the courses in 3rd year, though not always.