How important is major in the UK?

Longer answer to your short question:

1. Majors don't exist in the UK. Seems like a nuance, but it isn't. A philosophy student in the US at a top school gunning for IB is likely to have some coding/data skills, whereas that's unusual for it in the UK.
2. As a general rule, subjects matter somewhat but probably less than the US. There's broadly a top category of STEM/Economics followed by Finance/Management and then other respectable non-quant subjects (History, English, Philosophy, etc.). Getting into FO out of subjects with less transferrable skills - think Classics, Modern Languages, etc. - is near-impossible outside of Oxbridge. 
3. UCL is a target, albeit a lower one, and Philosophy is probably more respected in the UK than in the US because of the dominance of PPE in public life. I agree with the post above that UCL philosophy is better than most semi or non-targets in something relevant. Would probably say it would be comparable to STEM/Econ/Finance at an upper semi-target like Edinburgh or Durham. You'll have a chance - but to compete against STEM/Econ/Finance students (a large proportion of whom won't get FO from UCL) you'll want to get Spring Weeks. Would also advise that you try to take more quantitative modules/develop decent programming skills to tackle the "social science stigma" that exists in the UK. 
4. Finally, UK targets - even doing Econ/STEM - are not guarantees of FO. The lack of proper OCR means that UK targets are nowhere near as strong at placing than the top US universities, and you also have very experienced continental Europeans that you're competing with. Look at LSE's placement if you don't believe me - less than 15% of graduates land FO jobs. Situation is even worse for UCL, as ceteris paribus you've got two London unis (LSE/Imperial) that will be preferred plus Oxbridge too. 
5. Assuming you're American - based on your use of the word "major" - I would strongly consider any US T20 target over UCL. Per above, recruiting is easier, and student resources will be much better.