Umm Tolkien would not have regarded himselfas a celebrity and in the UK celebrities are not really involved in party politics (as opposed to issue politics) though our most recent former PM was a bit of a fame junky .... so objectively -
Tolkien seems on the surface to have been what we call here a "small c" conservative. However I am sure his childhood experiences would have made him in favour of the NHS and the safety net provided by the welfare system. His treatment of Sam might be seen as an endorsement of social mobility - a move away from the hereditary principle towards meritocracy. He restores the king it is true but that King devolves power in the Shire to it's inhabitants. As a Catholic he may also have had a sympathy for those who were outside the establishment.
We can fairly safely assume that ecological issues would have been extremely important to him - he wrote "not a penny for concorde" on his tax payment and we know he would have had a very small carbon foot print. He owned a car only briefly, travelled abroad little and walked and cycled for day to day transport. His homes had very little in the way of mondern conveniences (apart from the few years in Bournemouth) - an example to us all in that regard.
He was a soldier and participated in the battle of the Somme in which a million combatants lost their lives. His letters reveal his anxiety about his sons who served in WW2 so neither "surrender monkey" - to use a horrid phrase - nor war-monger I would judge. Certainly he would admire McCain's courage and powers of endurance shown during his time as a POW regardless of what he thought of his politics.
I am not qualified to judge which of the candidates he would prefer and certainly I doubt he would do anythong so vulgar as endorse anyone - I have to say that having been raised to respect the secrecy of the ballot (though ballot papers are numbered and could in theory be traced) - the idea of even being registered as a voter to a particular party is somewhat bizarre.
However although Tolkien was born and died in the reigns of two great queens his works would suggest that he did not favour "ruling queens"