One of the recent news stories that you may have missed is the idea of a Police Federation style "union" for rank-and-file members of the armed forces. A real union (with the power to organise industrial action) would, of course, be illegal under the Queen's regulations.
However, I feel that a real, TUC affiliated, trade union would be entirely appropriate for members of the armed forces. They are expected to disobey orders that controvene the Geneva Convention and to stand up to the parade ground bullies but don't have the support of any independent organisation that can represent them. Indeed, I can think of no circumstance that could warrant industrial action more than an officer (or politician) ordering soldiers, sailors or airmen to do something that would clearly constitute a war crime.
Indeed, it's arguable that British soldiers invented modern trade unions when rank-and-file members of the New Model Army elected "agitators" to represent their views at Army Council. These agitators went on to become powerful advocates for greater democracy in the country, especially at the famous "Putney debates". Maybe that's why John Reid is so cool on the idea.