Alnwick. 3rd Weds The Fleece - opposite the Playhouse. Session (tunes - very few if any songs I would imagine)
Berwick. The Pilot, Thursdays, I think. Session as above.

There used to be a folk club in Rennington - on Sundays?, haven't heard anything of it for ages though.

Shilbottle (just south of Alnwick) Pipers session (ie in F pitch,) 4th Weds of the month (24th in August).

Can't help with north of the border.

There's also fiddle and accordion clubs in Wooler and Kelso, but I don't think they're participatory.

There's another group of sessions in Amble and thereabouts but I have had very mixed reports - some think they're fine, some say Oirish thrash (which is not a compliment).I've never been so 3rd hand info.

Hope this helps
Paws