I can't really make any recommendations for London specifically with a little one, we haven't taken our almost 5 year old daughter there yet, but I will say don't be afraid to take them. Yes you will have to adjust how you travel since you will have a third person that relies on your for everything. You probably won't be able to do everything you want but you will still enjoy what you do and make lots of memories. Plus you can see things through your daughters eyes, which might change how you look at something. Our daughters very first trip was at 9 months old to Manchester, England and Munich and she has returned to Europe every year since. People thought we were crazy for traveling with such a young one, and some would probably still agree, but we love to travel and wanted to instill that in her. She's an excellent traveler, it may be luck but I like to think it's because we just jumped in and did it (or because I went to Germany while 5 months pregnant with her) and she loves to fly.
Plan to adjust your day based on if you need to return to the room for her to nap. Our daughter napped in her stroller with no problem so there were many days she would just do it that way, other days I would go back to the room for her to nap in the room. I never had a problem finding diapers, just know her weight in kg (pounds not always on the package), as the size number is a little different than what we have in the States, they even have Pampers if that's what you prefer. I've never had a problem with getting a crib while in Europe, sometimes a pack n play but just know that a lot of hotels will charge for that crib/pnp per night. Our hotel in Munich usually charged 10-15Euros per night. Try finding a park one day or two to allow her to run and just play. I know some people don't like using them but at that age we carried one of those backpack harnesses with us so that she could walk. She was holding our hand most of the time but that was also around the time that she was resisting that so it helped us to have it. She actually loved wearing it because it was a monkey backpack.
In my opinion you can always find something that a little one will like to do but at her age she'll most likely go along with whatever you are doing with a few minor adjustments. At almost 2 they are more interested in playing with toys or playing outside and that's something you can easily accomodate just about anywhere.
I can easily give you other tips and advice. We have been all over Germany, to Manchester and Copenhagen with our daughter so I've learned a lot of things. As long as there is money to go overseas we will always do it at any age with her. Don't be afraid of it.