Carb,
Yes, they are allowed to say No, and yes, some of them frequently do. Local knowledge usually helps decide which airspace it's worth asking for a transit through and which isn't.
As a generally rule, the larger civillian airport don't like it, whereas the military guys and the smaller airfields are quite happy to let us through. Of course, there are always exceptions to this. I understand that Thames (who control the area around City airport) will let you through their zone quite happily - although I don't think most single-engined aircraft could comply with the rule about being able to land clear. I've certainly found Thames very helpful when getting a flight information service outside their airspace.
Having done some flying in the US, I agree that the attitude there is very different. I loved doing the Class B transition at Phoenix - can't imagine anyone ever let me fly that close to Heathrow in a Super Cub!
FFF
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