I guess it really comes down to your personal interests, personally I don't think your itinerary is that bad unlike the above posts.
On the 2nd day, I'm assuming you'd be planning on doing the London Eye in the late/afternoon/evening after visiting Westminster Abbey before wandering over to Trafalgar Square and through to Covent Garden for something to eat afterwards.
The weather is forcast to be below zero in the evenings until the new year and I can't think of such a contrast between hot, humid Singapore and a chilly evening in a London winter.
I'm assuming, you're visiting the Tate Modern and not Tate Britain.
Regards Changing of The Guard, my impression is that it is one of the biggest anti-climaxes of most visitors, especially the one at Buckingham Palace. The one at Windsor however I find OK (as part of a visit to Windsor Castle) and if you get into Windsor on opening its timing is quite convenient after you've visited the core of what to see at Windsor on a take it or leave it basis. Relative lack of crowds mean you can dip in and out, I think it lasts about 30 mins and St Georges Chapel is adjacent to where it takes place within the castle..
Not sure what your expectations are with Buckingham Palace and Trafalgar Square. Buckingham Palace I guess is just a 1 minute photo stop you might grab on your other days..
Trafalgar Square itself is just a traffic island where people hang out, though your interest in the Tate may mean a visit to the National Gallery/National Portrait Gallery on the northern edge may be of interest, but time is very much against you and there is no late night opening on your dates.