The UK national route planner can't find any sensible bus or coach routes for your journey:
http://www.transportdirect.info/
(The only way seems to be via London, with many changes and taking over 18 hours)
National Rail can get you to Alnmouth, which is about 3 or 4 miles from Boulmer, changing at Manchester Oxford Road and York, in about 6� hours:
http://www.nationalrail.co.uk/
The single fare is �70.40. A return ticket costs �110.30 with no restrictions on travel times, or �90.40 for off-peak travel.
There appear to be no bus services, other than at school and college times, between Alnmouth and Boulmer, and there's no taxi rank at Alnmouth station. Phone Alnmouth Taxis on 01665 830096. (Alternative firms: AA Alnwick 01665 606060 or ABC 01665 603000).
Chris