We're spending four nights in Rome in mid-May and I agree that it's REALLY tough to find anything affordable. I used a site cross-pollinate.com. They do accommodations only for Rome, Florence and Venice. They're very friendly and helpful even writing out specific directions from the airport to our apartment. Here's what we're getting: a studio apt with a balcony, private bath, elevator, double bed and twin sleep sofa (there are 3 of us), 5 minutes from St. Peter's (not central for everything but that's alright for us). It's 110 Euros/night which I don't consider cheap but I wasn't finding anything better. June's got to be just as bad. I looked at the photos and then emailed asking for advice as our 80 year old mother is going with us. They were so kind.
I also contacted goporta.com and got reservations for the Vatican Museum. It's just a small booking fee and the tour gets paid for when actually arrive at the museum. The advantage is that you don't have to stand in line for hours. The particular tour we're going on has an English-speaking guide along with headphones; it last two hours but you can stay over afterwards.
I hope some of the above is helpful. Although I LOVE all the planning, trying to find places in Rome and London that didn't break the bank was a HUGE challenge. I spent hours and days. Good luck and have a great holiday.
Yes, hotels in Rome AND London are really expensive. $400 and up a day isn't unusual. I've been telling everybody to go on hotels.com. Their list of hotels and prices is seemingly endless. Also check priceline.com, Expedia, Travelocity, and Tripadvisor. You certainly should be able to find something in one of those!