It's just not true - Chemistry at A-Level simply is not a requirement for the vast majority of Chemical engineering courses. Mathematics definitely is, plus a science from physics or chemistry, in 90% of cases for the top universities. For any engineering subject, maths = required. A single physical science (i.e. chem or physics) = usually required. Further maths = strongly recommended.
They're all similar, yet different. For starters, your first two years will, in all likelyhood, not differ greatly between all 3 courses - there will be plenty of maths, probably subjects like thermodynamics, and various other core lectures which will be similar. There will be some specifics on your specialism - but they will account for a few modules per year at best.
I studied electronics. I know about mechatronics, and I currently work in the Aerospace sector, so I can provide guidance on those ones. People will say some are harder to get into, some are not - your only real guide here (in that they're all engineering courses and will be hard to get into) is that the higher the typical grades required or offers, generally, this is because they are the most popular courses as it helps to filter out the "best" candidates.
So Mechatronics, or mechanical + electronics engineering. This tends to be a mish-mash of both the mechanical world and the electronics side of things. After doing this course, you'll be good at both, but your specialisation will determine which way you lean. People often equate mechatronics with robotics, which, yeah, I can see why - they're ideally placed for this (and indeed, there are a lot of jobs out there concerning robotics - 6000 openings on the IEEE website currently) But in reality, you're not at all limited to robotics - you can equally as well go into mobile phone communications or manufacturing, or any other job that lends itself to either type of engineer. Engineering is highly transferable between disciplines - in electronics for instance, the jobs that exist today didn't exist 4 years ago, yet are only open to graduates who *could not* have studied them. This is because an engineering degree provides you with the foundations to enable you to teach yourself, together with an understanding of applied mathematics and physics (even in chemical engineering) - so I wouldn't really worry too much about which specialisation you go into.
Aerospace engineers are essentially mechanical engineers, they just deal with fluids more, and what happens when things break phenomena like the sound barrier. Electronics engineers too, are mechanical engineers on the micro and nanoscale, although they're better at programming. Mechatronics introduces you to systems engineering - making things work together, no matter from what discipline they're from, and Chemical - well thats just dealing with the physics of materials and compounds. There is a lot less chemistry in chemical engineering than you're average A-Level student would believe - which is reflected in the fact that it's entirely possible to do it without having done A-Level chemistry. Those who have done it will find it relevant, sure.
Downsides to them? really not too many. Mechatronics is still a new-fangled combination, and the fact it doesn't explicitly say engineering can unfortunately count against you in job applications (easily rectified - put mechanical engineering with electronics in the form instead - helps the poor Human Resources people understand that yes, you are qualified to do the job) Aerospace sounds flashy, but "more opportunities than other engineers?" disagree with that statement - I work in aerospace, and less than half my colleagues of my age have aerospace degrees - as with most engineering, it doesn't really disadvantage you in terms of which area you want to end up in. I did electronics - didn't even study thermodynamics but I have the same problems to tackle as those who did. Chemical engineering is great if you want the highest paid *engineering* jobs in oil & gas, but if you're just out to make money, study engineering and go into finance. Providing the world economy doesn't collapse whilst you're at university...
Anyway, hopefully there are a few useful snippets in there for you,
best of luck with whatever you go for, oh and for what it's worth, further maths is your most useful A-Level, even if it's difficult and might drag your average marks down.
Stu Haynes, MEng, MIET, MIEEE