your pickings are somewhat slim with any sensible definition of "top 10".
I don't think someone looking for a specifically maths course should go to LSE, it really seemed like a stats & econ course with some pure on the side. It is more appropriately viewed as a competitor to say MORSE at Warwick. I don't know anything about LSE admissions but if they state that FM is required it's most likely they will just reject you off the bat. MORSE might accept you if you get a 2 in STEP or a good TMUA result, but they say this is for people who couldn't take FM. (you could ask about this)
Good maths universities that don't need FM include Birmingham, Lancaster, Glasgow, Edinburgh, Leeds. These are not really "top tier" per se but very very strong "second tier" and plenty go on from these universities to postgraduate study at "top tier" universities. (I know nothing about the Nottingham course but I know someone who went on to do a PhD at an Ivy League university after Notts undergrad and some others from there who have gone on to do PhDs at Warwick and Oxbridge, so these universities are far from something to turn your nose at. Several people from KCL went on to do Part III) Not to say you'll want to do PG study, but it's indicative of the course breadth/depth at the very least.
If you can do an AS in Further Maths this would open up universities like Bath.
Don't fuss whether unis are RG or not, Lancaster and Bath are both better than eg Liverpool, Sheffield, QMUL.