luggage

I have not done this tour but I have done other city tours. Rome and Florence.

I use a messenger style bag that I can keep in front of me for
guidebook
camera and accessories,
cash and credit card,
cereal bars,
Brita filter water bottle,
jacket or rain jacket or hat as needed.
smartphone. I'm starting to rethink the messenger bag as it is causes a little shoulder soreness after several days. I'm thinking that a backpack would distribute the weight over both shoulders but it brings up security issues.

As far as luggage, I now bring a 24" eagle creek rolling bag, but that's for longer trips and I don't pack light and I check luggage. For the city tour I doubt that you will need to change hotels, so take whatever you need for the trip (I'm assuming you're just taking this tour and do not have additional travel plans).

I have one large, 22" carry-on, wheeled bag and one small day pack. I always leave the large bag in my hotel room and carry the small day bag around with me when sightseeing to hold my camera, guidebook, umbrella or rain jacket and snack.

You should be able to fit everything you need in a 22 inch carry-on sized suitcase and one "personal" item. You don't provide too much detail (when, escorted or not, location of lodging, etc.) but even if you can't fit fresh cloths for every day, there are plenty of laundromats with drop-off service.

Not having to check luggage on a flight is such a relief!

I am guessing you mean the RS London tour. In that event you are staying in the same hotel the whole time so you just have to manage your luggage to/from the hotel. I would still go no bigger than 22" but that's just me. The hotel rooms may be a lot smaller than you are used to, so not as much room for luggage.

As far as a day bag, something to keep a rain layer in but small enough that you don't need to check it in some of the venues.