Museum Reservations

The British Museum is free( there was a place to put donations if you so desired though), so no real lines that I recall. ( been there twice, March and July )
I have been to the Louvre at least 6 or 7 times. Always in summer, busy season so to speak. I enter via the underground shopping mall , or metro station level, buy my ticket from the machines, and walk in. No real lines. Still a line to go through security, but no pass nor reservation will avoid that, and it only takes a few minutes.
Tower of London , well I have only been twice, once in March and once in July. Both times I waited about 15 minutes.

Vatican and Colosseum I saw this summer on a RS tour and we did in fact have group reservations, I can't speak to what it would be like off season though.

My opinions quickly. Most of these I will say you can walk up to and just buy your tickets. I think some forethought will always help for larger tourist spots like showing up at opening or towards closing.

That said

Tower of London - Walk up almost anytime / No Reservation.

British Musuem - Walk up almost anytime / No Reservation

Louve - Walk up, Good to go non peak hours

Uffizi - Reservation makes life much easier

Accademia - Reservation makes life much easier

Vatican Museum - Unless with tour group don't believe you can make individual reservation. Line up early. I think I have seen other posts here that just after lunch is good time.

St Peters - Unless with tour group don't think you can make individual reservation. going first thing in morning helps.

Pantheon - Just walk on in when open.

Colosseum - Unless with tour don't think you can make individual reservation. Lining up early helps. I think there is a tip on here to buy a combo ticket at Palentine Hill and use it at Colosseum.

Have not done London Eye or Scrovengi. Lots of Duomo's out. If you are speaking of the Florence one I didn't go in but for all the ones I went in I was able to walk up to and within 15 minutes get inside.

Hope that helps

Just so you're aware, the Christmas holiday season is only slightly less busy than the peak summer weeks.

Can't help you with Italy, but in Paris, if you visit more than a few of the major tourist sites, consider buying the Museum Pass. You'll save a little bit of money, but the biggest saver will be in time- at the major museums, the longest line is not for entry, but to buy tickets. With the Museum Pass, you can by-pass the ticket line. At the Louvre during the holidays, this can save you an hour or more. If you decide to buy the pass, buy it when you get there.

Tom, you do not need to line up for an hour at the Louvre, seriously, I have been at least 6-7 times now( always during summer) , and if you enter through the shopping mall and buy tickets from machine is will be walk in easy. Have you tried this entrance yet..? Everyone runs to Pyramid entrance, but it is the slowest way in.

Nope, I did not wait an hour, because I used the Museum Pass and walked right in -this was the day after Christmas last time I visited. But the line going into the pyramid to the ticket counter was VERY long. Same thing at the Orsay, and the Pompidou (that line stretched around the block). The whole point of the story is that the Museum Pass is a great time-saver.