I would guess that ALL British libraries are the same but can only speak for Gloucestershire. .
No. My local library, Stevenage provides free WiFi access.
And as I said earlier, West Sussex libraries charge for all internet access. You can buy a years pass for £12 otherwise you pay a day or half day rate (which quickly goes over the £12). One of the libraries has BT Openzone and some of the others may get some form of WiFi - but would still be chargeable at the aforementioned rate.
It is very different across the country as there is no government requirement to provide net access in libraries and the previous funding that used to provide it ended a few years back.