Interestingly, there are suggestions that the man who told Wilson about Mountbatten's plotting was...er... Mountbatten.
In the authorised biog of Wilson, Philip Ziegler (also authorised biographer of Mountbatten) gives some outline of what happened (at least in regard to the first alleged coup). Cecil King, of the Daily Mirror, had a meeting with Mountbatten in which Mountbatten got the distinct impression that King was advocating that he should lead a military coup to depose Wilson.
Mountbatten was sufficiently concerned at King's inferences to make a record of their conversation, which he agreed with King's deputy. He then went and told Wilson. Ziegler suggests that Wilson became rather confused as to what Mountbatten was up to and began to suspect Mountbatten.
It'll be interesting to see whether the programme clears this up or not.