I suppose I'd go with a full length, wool topcoat, in grey.
The rationale being, a grey topcoat is a perfectly fine choice with a suit, but even when wearing relatively casual clothes, it won't look hideous or anything. Whereas an informal jacket might be a good choice with casual clothes, but will tend look quite wrong over more formal clothing.
In other words, better to pair a more formal coat with sportswear, than an informal jacket with a suit.
As for grey... well, it's grey. It'll go with most colors. Not always brilliantly, but acceptably.
Wool, because it's a classic, obvious, genuinely good choice. Readily available and often reasonably priced. Sure, 100% cashmere (which is technically a type of wool, of course, but I trust that when I speak of a wool coat, and don't specify cashmere, it's understood that I don't mean cashmere) can be great - particularly high quality cashmere - but it's also likely much pricier than the non-cashmere route, and since you specify that this coat will only be need for a few months and you want to spend much...
If you're going to be doing a lot of driving, maybe get 3/4 length, instead. Maybe. Although I drive a lot, and I don't find full length to be a big problem. And I just think that full length looks better (more classically styled, if you will) and certainly it provides more protection from the cold (which I'm thinking could be important to someone coming from a very warm country, who probably isn't used to cold winter weather).
I prefer traditional fit, to slim fit, and particularly prefer it if one is going to be layering (which is a solid idea, in cold weather). But since I know nothing of your age, build, sartorial preferences, etc., I'll refrain from telling you that one type of fit is the right choice, and the other the wrong choice.
I don't know precisely what's available in London, so I can't recommend a specific store. Here in the US, I'd tell you to pick up a $129 wool overcoat from JAB, and call it done. It wouldn't be great, but it'd be most acceptable, given the situation you've described.
Me? I found an essentially unworn, high quality, full length, Brooks Bros., grey cashmere overcoat, on eBay, back over the summer. It cost me less than the sales tax on such a garment, purchased new. Overcoats are one of those items - like blazers - which seem to be plentiful on the used market, often selling for pennies on the dollar. But since it's already January, I'm guessing you need this coat ASAP, and preferably with a minimum of fuss and bother in terms of purchase, so this route likely isn't right for you.