Avoid Heathrow if you can. The airport charges onerous landing fees. Using Avios miles on British Air metal will pass on high fuel surcharges.
United --
Chase United x2
Chase Sapphire Preferred with transfer to United x2
AA --
Citi AA x2
Barclays x2 (1st year isn't waived)
Looking for award flights in July may be too late, though, without paying "full price." That is, they release only a few "saver" awards per flight and they go quickly. You may have to break your flight into 2 one way trips -- use miles for the flight over and pay for the return flight.
Also, when looking for award flights, try taking out the first hop. For example, look for an award from EWR->HAM rather than a flight from DEN->EWR->HAM, for example. The domestic flight has limited award seats, too, and they go more quickly. You could then use SWA miles to get to EWR.
Finally, a 24 hour layover might be ok if you get a hotel. Reykjavik isn't a bad place to look around town and get adjusted to the time change. Think of it as part of your trip.