As Wiremill says the Tyne is a short drive from North Berwick - the water is extremely low at the moment but you will get easy access and a day ticket (£7)can be got from the sport shop at the west end of the main street in North Berwick (forgotten the name).
Pretty much the whole river is available under the ticket (see
ELAA Home Page) and there are lots of sea trout in at the moment.
A short rod about 7-8' will be more than enough and you will find a lot of the river is overgrown.
Spots I would recommend are Hailes Castle, Abbeymill Bridge (go downstream) or Sandy's Mill.