Scheduled Flights from Manchester: (None to Dortmund. None to Weeze. None to Paderborn.)
BEST OPTIONS:
Manchester - Dusseldorf: Lufthansa & FlyBe
Manchester - Koln-Bonn: Germanwings
When the above get too expensive, consider:
Manchester - Frankfurt Main: Lufthansa
Manchester - Hahn: Ryanair
Good rail services from Frankfurt Main; Hahn is a coach hub.
If desperate, decent options further afield:
Manchester - Amsterdam: EasyJet (cheap) plus ICE3 rail connection. KLM also fly there but fares tend to be alot more expensive.
Manchester - Bremen is served by Ryanair. It is a long distance from Dortmund but the flight can be very cheap indeed and rail connections down to the Ruhr are good. May save money for a late booking.
In summary, use Dusseldorf or Koln-Bonn Airports if you can. But these will fill up quickly and become very expensive. If you are willing to consider a longer train / coach journey, the other listed airport options are viable as a 'Plan B'.
Enjoy the trip!