"Multidisciplinary degrees might be the way forward - a bit like in the US where you can major in, say, History but also take modules in chemistry, statistics, music etc."
You can do that at many universities in the UK. I frequently teach humanities modules to engineers and chemists and IT students. It's the kind of thing you should look for in prospectuses and ask about at Open Day.
"I have a first class engineering degree.
Useless. Absolutely bloody useless. Turns out to get a good job in engineering you have to do loads of work experience BEFORE you graduate. Stupid me.
Also, being brutally honest, I experienced massive sexism and some very uncomfortable borderline harassment trying to get in to the sector. Not nice.
I wish I'd not wasted so much time and energy trying to get in to a sector that clearly was never going to be the right one for me. But difficult to know what else I could have done with that time."
Apart from the sexism, which is clearly disgusting and not the responsibility of the new graduate, the other part (the lack of experience) is often more in the hands of the individual. This is something my generation was badly prepared for, but something I always try to suggest to new students: that they should spend part of their study time thinking about the future, take any careers advice on offer, getting relevant extra experience, not leave thinking about a career until the last semester.