eBookers is one of the most long-established and well-respected firms in the UK travel industry. You do need to take care to check what it is you're actually booking, but otherwise there should be no problems. (For example, if you're offered a package holiday then you'll get all of your money back if the airline goes bust. But if you're offered a 'flight + hotel' deal, and the airline goes bust, then you'll still be contracted to pay for the hotel).
One of the best things about eBookers is that you can talk to a real person (at standard phone rates) 24 hours per day. Call 0203 320 3320. (New bookings are only accepted between 0730 and 2345 daily).
However you might still do better to book your flight and accommodation yourself. There's plenty of competition on UK to Rome routes, so fares are usually quite low. (Ryanair, in particular, has flights from lots of UK airports, into Rome Ciampino, which is closer to the city centre than Rome Fiumicino which other airlines use).
Chris