biggleswade to wooler best way to travel bus or train?

Best... do you mean quickest? Or cheapest?

You'll need to catch a train from Biggleswade to Berwick Upon Tweed (change at Peterborough). From Berwick Upon Tweed (Golden Square)there is a regular bus service to Wooler. Book as far ahead as possible for the cheapest rail fares either at your local station ticket office, or at www.nationalexpresseastcoast.com
For local bus info in North East England see www.traveline.info