The HO/HO buses in London aren't recommended because of severe traffic congestion. I don't know about Dublin and Budapest. I'm a fan of walking, and Budapest's beautiful architecture makes walking especially attractive there.
I've never been to Ireland, but I think a week would be a very short trip to Hungary. Not bad if you think of it as a trip to Budapest with one day trip. Budapest is especially rich in sites related to 20th-century history. My problem is that I really liked seeing Pecs, Kecskemet and Szeged, and I wouldn't recommend that much travel on a one-week visit.
The travel time to Hungary (at least one connection required) would probably be longer than to London or Ireland, and the flights would likely be more expensive. But hotels and good food would be a lot cheaper than in London and Dublin. I don't know about costs elsewhere in Ireland.
You might take a look at hotel rates on booking.com. if there's a big spring-break bump, hotel costs in London and Dublin might put some serious pressure on your budget. I've used the Vincent House in London as a less-expensive option on solo trips there outside of air-conditionjng season; VH doesn't have a/c.
A plus for London is the huge slate of £15 walking tours offered by LondonWalks. The company's guides are licensed, and there are tours on a large variety of topics--some are historical, some are neighborhood-focused. The other positive for London is the extremely large number of large museums that are free/donation-requested. There are some quite expensive sights, including Westminster Abbey, the Tower of London, The Churchill War Rooms and St. Paul's Cathedral, but you could easily spend a week just in the free/donation museums like the British Museum, Imperial War Museum, Victoria and Alnert Museum (decorative art), National Gallery and Tate Modern. (Those are just some of the options without admission fees.)
