We are talking "Newcastle Upon Tyne" - and not one of the other Newcastles???
The "Name" river in the area is the Tweed - lower beats, even in July, are spendy and the main run of fish on the east coast rivers is usually late October onwards....
But within a couple of hours drive of Newcastle you have:
Annan/Nith/BorderEsk/Liddle - All of which are Scottish rivers draining into the Solway.
The Eden and Lune in Cumbria - Ditto but English
The Till/Teviot/Leader/Ettrick - these are first tributaries of the tweed and available quite cheaply in rather more senic surroundings - can fish for salmon but unlikely to see one in July.
The Kale/Ale/Jed/Rule etc - these are second tributaries of the Tweed - so no Salmon fishing but wild brown trout in smallish streams - not easy but great fun.
Links
Fishtweed
Jedforest Angling
Hawick Angling Club
Kelso
You could google for Earlston, Selkirk, Galasheils and Peebles each of which has its own club or association.