How much adventure do you want?
I would say your top choices are:
Manuel Antonio - beach town on the edge of a protected forest ("national park") which contains 5 beaches and jungle that you can walk through, absolutely teeming with monkeys and birds so animal spotting is very easy. The protected area is totally unspoiled, there are no restaurants, toilets, shops inside the area, just jungle and beach. The town on the edge of it is quite built up though (by CR standards)
Monte Verde cloud forest. Lots and lots of different ways to spot birds here, there are tree top rope bridge walks, ariel trams, and zip lines. Great if your kids are into adventure type stuff. Some interesting accomodation in the form of tree houses and mountain lodges too. Slightly damp most of the time so not as "tropical" as a beach trip to Manuel Antonio.
Tortuguero is also amazing for animal watching, people mainly go to see the huge turtles nesting there (check the time of year for this) however you might find the journey arduous, its a long drive and then a long river boat ride from the airport, might be too much after such a long journey from the UK.
You can also organise tours pretty easily so that you can spend a couple of days in each of these places (these would normally include a trip to see Arenal Volcano too - fantastic seeing it explode lava at night from the comfort of a hot spring swimming pool with swim up bar!)
My friend runs a very nice very reasonably priced hotel in San Jose if that is where you are planning on flying in to. They have a tour desk too, PM me if you want details.