Thanks for the great info, Chloemom. 

We'll be traveling through Heathrow later this spring with nut and shellfish allergies as well-- luckily not the same person, though.

I'd check with the airline.
It depends on which terminal you'll be in as to where you eat but the UK have pretty strict guidelines about labelling, places like pizza express and pizza hut have allergy labels on their menus. Otherwise you could get takeaway food, sandwiches, pasta salads etc from places like Boots and WHSmith. I'd be wary of coffee places as they may claim to be nut free but often have things out in the open and serve almond croissants next to regular ones.

Has anyone flown British Airways?  I read on their website that they don't serve meals containing peanuts (but have tree nuts.)  Are there any safe places to eat at the Heathrow airport terminal for someone with nut and shellfish allergies?  Thanks!