The plumber trade is associated with Field Construction Squadrons within the
Royal Engineers rather than REME. Trade title is 'Military Engineer (Heating and Plumbing)'
This site gives further details regarding initial training, further training, workshop skills etc. Clerk of Works (Mechanical) would be intended final trade training:
http://www.armyjobs.mod.uk/Jobs/Pages/J ... category=2
If you're already a qualified tradesman you will certainly have an advantage, but bear in mind that there are in truth quite a few applicants in a similar position to yourself, especially in the current climate of the housing construction trades being hit during the recession.
As is true with ANY trade in the Army, you are a
soldier first, tradesman second so will still have to do your Combat Engineer training and other military courses.
Give your Careers Office a call.