We had really good luck on our previous two trips to Europe using miles. But for our trip in September it's been a challenge, especially since we wanted to fly business using saver miles. After changing the economy flights 4 times, I finally snagged a pair of business seats.
But here are some tips I have based on experience
1) Do you have silver status with United? If so you can change your dates/route on award tickets as many times as you want free as long as the origin and arrival cities remain unchanged. If this is the case you could book your three outbound tickets now, then in two weeks look for your return options. If you need to tweak your dates, you can search both one ways based on availability at both ends. Then I'd call United and explain what you want to change on the out bounds and book for the returns. I've found the awards customer service people are very helpful.
2) I believe United hubs out of Newark, so you may have more possibilities searching out of that airport than JFK
3) For the 3 fares you are purchasing, you have to book round trip as one ways to Europe are more expensive than RT. If I recall when we booked our flights as soon as flights opened up, they were more expensive than waiting a few months. Maybe search flights 2,4,6 and 8 months out to see how the prices change as you get closer to your travel dates. Flights won't sell out until you are really close to travel dates,
4) If you want to book all the flights together, definitely call instead of booking on line.
5) Once you have all six tickets booked, call United and ask them to connect all the reservations. That way if there is an itinerary change, or rerouting, you will all me moved together.
6) Search for the US to Europe flight first- JFK - LHR or EWR to LHR or IAD to LHR or FRA to EWR or FRA to IAD, etc. Figure out which routes, flights, might have direct flight availability. So if you see FRA to EWR availability, then try (FCO) to EWR and see if anything comes up. I've been able to put a saver miles flight together by doing multi-stop saver flights that didn't pull up on the one way search. I tried SFO to LHR and nothing, but then I tried SFO to FRA and FRA to LHR and bingo found saver availability (connection in Charlotte).
7) check your flights regularly for changes. If departure time or arrive times change drastically, or if your connection time changes (especially if connection is really tight), call United to complain, but have your ducks in a row with the flight options you want (another reason to connect all reservations). For our flight back from London in September, we had a flight from LHR to IAD to SFO leaving London at 4pm. Welll United cancelled the flight from DC and put us on a noon flight out of London. Called United and explained that we were taking a train from Edinburgh(we weren't) and the noon flight wouldn't work. I suggested the direct flight from LHR to SFO at 2PM, and without question, they booked us on the direct flight, even though there were no saver flights available.
Hope this helps