Unless you've already been barred is there a Public Library anywhere near you?
Dry, warm & generally quiet.
PC stations plus selection of papers - Local & National
Some have a cafe, most have WCs.
Librarians knowledgable, (some in both senses of the word, nudge nudge wink wink)
Alcohol free - unless you BYO in a discrete brown paper bag a la USA
Brother on a day like today, if I had my offspring with me and I saw you trying to avoid the public... I would go out of my way to sit next to you and give my youngest fucking pixie sticks just to be that kind of asshole.
You get used to it.
Sorry Red, but as a newly badged BRIT, you'll have lived here long enough to have had a serious amount of advise....... SO WHY DID YOU IGNORE IT!!!!!!! Next time a Limey tells you to stay away from London listen to them