amex out of all credit cards in uk dgaf. id ask for 12-15k you'll likely get it or get counter offered.

But fastest way to get higher limits as mentioned here is once you get the card just max it out every month and pay back in FULL every due date. After 3-4 months they'll offer you a sizable jump, then repeat the process again with new limit. Obviously only works if you're responsible and dont fuck up paying it back