No UK university is really comparable to a US Ivy League institution for teaching really because of the nature of the very different system which prizes independent learning over classes. The Ivy Leagues aren't the top schools in the US btw; they stopped being about sports a long time ago.
In terms of brand, the schoolchildren here aren't representative of employers in the UK in general. On both sides of the Atlantic, an Ivy League uni would provide a stronger brand name/better recognition for large companies than anything in the UK, except for Oxford and Cambridge. Oxbridge probably has a better reputation than most of the Ivy League. LSE and Imperial *might* be able to compete on brand name, in some countries more than others; but not necessarily.
In the case of LSE, it should be emphasised that it is really a non-institution. It is a living brand name. It's a marketing company, more than a real university. The student experience there is one of the worst at UK unis. It's a small, cramp campus often with foreign lecturers who don't give two ****s and the limited contact time with them is just the same as at other UK unis. It takes rich internationals, postgrads; charges them a truckload of money, and that's why it's a comparatively rich small institution which can afford to spend millions on self-promotion all over the world.
Most UK unis are good and we have a lot of them, but the American unis are simply much better and if you have the money you should definitely go. But the UK, even with the fee hike, offers a much cheaper alternative. It only competes on price, not student experience and all the rest of it.
Looking at your post, it seems you're from HK and had a choice between Imperial and LSE. If you had been that specific, from what I know about HK; both those unis are very well known and respected out there. And would probably look just as good as a US university. But it depends what you want to go into. I said the above generally. For instance, I don't reckon many American employers in New York really know about Bath uni.