Hi - we have 3 kids (6 yrs old and 3 yrs old) and live in Yorkshire. We usually try and book our trips ourselves, using low cost airlines and searching for our own self-catering accommodation. We have had some great, inexpensive trips, which we have all really enjoyed.
If you want a short break or a week in Kent/Sussex (beautiful areas, very close to London, but a million miles away in many repsescts) - can I recommend you look at
www.freedomholidayhomes.co.uk - we just stayed in property ref. CB545 over Easter - it was wonderful - very nice farmer and family who own the property.
We have visited Paris a couple of times with the kids and cannot recommend enough
www.citadines.com - self-catering apartments that you book by the night - not the most romantic of buildings or locations, but great for families. Take the Eurostar - about 100x times better than flying (kids or no kids).
We have also rented houses in France - try
www.cheznous.co.uk - you book through the website, but directly from the owner - we have always really enjoyed these trips.
This summer, we're off to Denmark - a few nights in an apartment in Copenhagen and then a week in a beach house.
Gosh, don't we sound like a bunch of gadabouts!
Anyway, make the most of your time here to discover Europe, use those generous UK holiday allowances, and search around for some good deals. Also, don't overlook a holiday in the UK - they may just change your view of the country if you're used to London life.