Is flying out of Scotland an option? We always had very good luck with flying out of either Edinburgh or Glasgow Airports. Nowhere near the crowds of Heathrow. We flew Iceland Air a few times, and Aer Lingus (which puts you on a puddle jumper to Dublin to catch the transatlantic flight - watch the size of carryons!) was fine, too. We went well out of our way to avoid going through Heathrow just for that reason - the mass of people and the time constraints.
(And the fact that every time the Daughter went through there there was one hassle or another for her by the security people/TSA equivalents. One time she had to make a relatively tight connector to Glasgow and they pulled her out of the line because their sniffer had alerted to explosives. They were not very nice to her at all - she was in a wheelchair at the time, and they were very bullying. The airport helper who was wheeling her tried several times to intervene, asking them to remember proper manners/procedures. Didn't help. So after a West-Coast USA to London all-night flight, they gave her static about carrying explosives. Turned out to be her knitting yarn was setting it off. They they gave her a ration of BS about where had that yarn been.... Like at the yarn store, dude?)