I agree with the recommendations (if you haven't already been doing it) to spend some time checking out flights on Google flights, and also at the Trip Advisor air travel forum.
I live in Denver and have been envious of those low fares to the UK from East Coast airports, especially New York. I burned quite a bit of time researching how to take advantage of them. Besides the low fares, those flights have, for me, the advantage of being shorter by 2 or 3 hours than nonstops from DEN. After my research was done, I'm in agreement with what's already been said:
This might be a false economy as you will need to stay overnight to
avoid any mishaps. i.e. you will book on two different tickets and
have to pay for a hotel overnight in between the flights. You lose a
whole day and have to pay for the hotel. That said, cheapest flights
will be the ones with the most competition ...
and this:
There are too many potential problems and stresses involved in using a
separate ticket to reposition in the US on one ticket and
transatlantic on another ticket. Any mechanical, weather, air traffic
control problems with the first flight means you could miss the second
flight and lose the full amount of money you paid for it.
You minimize those issues if you don't travel all that far from home to get your transatlantic flight. For instance, my favorite two options from DEN to the UK are (1) the BA nonstop DEN-LHR, and (2) the Icelandair flights - they all connect through Reykjavik and you have choices besides London of your UK arrival/departure city (I've flown into Glasgow and home from London which was great for that trip). But last year a local friend got a deal for her family (I think on BA) via Salt Lake City, single ticketed and not a long connecting flight from here. So you just never know what's available until you do specific research.
Obviously it's hard to generalize about airfares, but I think you've gotten some great advice. Honestly, this city is getting more crowded and expensive all the time so I've considered moving. But I'm spoiled by having a major airport right here. Maybe someday when I'm no longer able to travel . . .