The Cambridge tripos system seems really good - broad first year etc.
I did my degree at Edinburgh, and although I signed up for Chem Eng, I did the same course as all the other Engineering students in the first year. As a Chem Eng student, I had to do Chemistry as my third subject, whereas other Engineering students had a fairly free choice. In my second year, there was a strong overlap with the Mech Eng students, then we were on our own for the last two years. This was over 20 years ago, I hasten to add.
It is good to be in a broad first year - firstly, you can swap to another discipline, but more importantly, in the world of work, you have to work in multidisciplinary teams, so it is useful to have an understanding of what everyone else is doing.
As for Further Maths, the Cambridge site says that it is very desireable, but recognise that not all schools offer it. If you can't do a full A-level, then as many modules as possible will make life easier. The maths burden for Chemical Engineers is immense, so I would definitely advocate doing Further if at all possible.
My DS is in lower 6th and doing both Maths and Further - he does not view them as double the work. I think he will be doing one full A-level this year, and the other next. He is also doing Physics, Chemistry and Philosophy - although still has no idea what he wants to do.