ISTR reading somewhere that Nelson (in his lifetime) pissed off a lot of people in London (not just at the Admiralty, but also in Whitehall and the East India Company).
The Admiralty is understandable/self-explanatory.
Whitehall is probably the result of the fact that First Sea Lord at the end of Nelson's life wasvSpencer, and IIRC he and Nelson didn't like one another much. Nelson had made some rather unflattering comments about Spencer*.
The East India Company/City Merchants were also opposed to him for some reason.
*Nelson, while much ink gets spilled on him about his genius and life, little gets said that he was genuinely not a very nice person. I'm not talking about his opposition to abolition of slavery, I'm talking about his actual personality. Several contemporaries - friend and enemy alike - all commented on it IIRC. ISTR of the people who personally didn't like him were William, duke of Clarence (future William IV) who had served with Nelson; and Maria Karoline of Austria, queen of Naples (she tolerated Nelson because she liked Emma Hamilton).
As for Nelson going into politics, he and Wellington apparently absolutely loathed one another (this was a personal animosity AIUI), so I suspect Nelson will wind up in the party opposite to Wellington's.
Another thing, regarding promotions, is that after Trafalgar, there were no further major major naval engagements (certainly none that would be defined as battles or worthy of promotion). A surviving Nelson will likely languish as Vice-Admiral of the White (a post he'd only held since April 1804) until he's at least 50 (1808). SUre, Trafalgar was a big win and all, but IIRC the British navy at the time ran on "seniority" and "patronage" (and Nelson having made the "enemies" he had (Clarence, Spencer, Wellington) had been a big reason why his promotion had been delayed previously. So unless he hits his head at Trafalgar and becomes a miraculously nicer person...think Nelson's future for the short term is pretty bleak.