Despite the presence of the word 'international' in their names, neither Bangor International Airport nor Portland International Jetport have any international services operating to or from them. This means that you'll have to change flights in the US. To see which cities have links into Bangor or Portland, check these links:
http://www.flybangor.com/airlines.html
http://gonewengland.about.com/gi/dynamic/offsi te.htm?zi=1/XJ&sdn=gonewengland&zu=http%3A%2F% 2Fwww.portlandjetport.org%2F
Your best option is probably to fly, with American Airlines, to Boston and then connect with a flight operated by their subsidiary, American Eagle, to Bangor. You can check the flights here:
http://www.aa.com
(Enter 'LHR' in the 'from' field. Put 'BGR' in the 'to' field).
On the random date I've just tried, the shortest total flight time is shown as 9 hours 45 minutes. However, there's a warning shown that the connection time is fairly short, at 1 hour 40 minutes. All other flights have total journey times between 13 hours 25 minutes and 16 hours 55 minutes. Times for return flights are shown as between 10 hours 15 minutes and 18 hours 0 minutes.
Chris