I know that the accepted advice on getting British money is to get it at an ATM at the airport on arrival. Sound advice in general, I agree. But on my last trip to the UK last week, I noticed that Terminal 4's so-called "cash machines" were (all but one) actually Travelex machines with their typically horrible exchange rates, not bank ATMs. There was a single bank ATM in Terminal 4 arrivals--an HSBC ones that was out of order. I know there used to be a Barclays machine there, but it's now gone. I checked Terminals 1 and 3 and it's the same situation there, too. I'm not sure about Terminal 5, but it may be the same there, as well.
So, since you are going to get the ripoff exchange rate at those machines, plus having to pay the cash machine fees from your home bank, I would recommend getting the smallest amount of pounds that will take you to your next destination at the live Travelex windows, then use a real bank-sponsored ATM in town to get your first major stash of cash. That way you minimize the ripoff rate exposure and will avoid the machine ATM fee in addition.
Of course, if you have a bit of sterling from an earlier trip, so much the better.