Well North Wales and Lakes both tick the boxes. North Wales has more trail centres and better access to the coast
I love Pembrokeshire bit its to far from much off road cycling
But I'll give you a pecific recomendation. Obviously one of millions of options but you've got to start some where.
I'd do a week in Elter Water in the lake district.
http://www.streetmap.co.uk/idld.srf?X=332905&Y=504890&A=Y&Z=120&lm=1
We've been every summer for about 15 years. Not close to the trail centres but not to far off either. Both Grizedale and Whinlatter have cafes and "go ape" as well. Its at the centre of lots of natural trails, its normal to see mountain bikers sat on the green having a mid ride break. There are loads of walks of all level of difficuilt nearby.
The Britannia inn does great food. You can walk to Chapel style to shop.
But the real ace is that if you pick the right cottage you'll get free country club membership for the Langdale estate. This means if it rains you can head off to the pool or gym. You can also head into amble side for a film
Its close enough to coniston that last summer I cycled there, mainly off road for a half days sailing (everyone else went by car). Lots of other water sports there as well
The only major down side is that the roads aren't ideal for cycling as a family. The road up and down the valley is a B road that just squeezes in 2 lanes. Its fine for an adult but not ideal for kids who are younger and wobblier.
If tell me how manny you'll have in the party I'll try and recomend a cottage