Hey @Seb888 - London is a true marvel, when it comes to anything drink related imho. You should definitely mark some time in your schedule for exciting bars and shops. 
Here are three places I would highly recommend to visit if you have a chance:
Trailer Happiness // https://trailerh.com/
I don’t need to praise this place for being a famous bar in the first place, I guess 
As a well known waterhole for those rum geeks among us, this is highly recommened. Very friendly and knowledgeable staff who are massiv geeks themselves and happy to engage in the nerdiest of talks, if time provides. Drinks are top notch. The rum selection is good but managable I would say. Some pics from my last visit below for reference:
Merchant House // https://merchanthouse.bar/
This is a stunning place, with an extensive collection of spirits. They offer countless bottles of whisky and also very nice drinks. One of their whisky cabinets for reference:
While their focus is on whisky, you will also find a very good selection of rums. This is a good place, if you are looking for the upper end botttlings (i.e. Velier Demeraras / Caronis / etc.)
Staff is also very friendly and knowledgeable, indeed. However, their “You will only buy what you like” policy knocks this place truely out of the park for me personally… before you buy a dram, they will pour you a (very) small sipp for you to get a first impression of the rum. Then you can decide to buy or not to buy a dram (obviously you will buy them, anyway ). This is true for all bottles they have.
Obviously, the exclusive bottles don’t come cheap - they price any (exclusive) dram based on current market value of respective bottles.
Black Parrot // https://blackparrotbar.co.uk/
Last but definitely “first” is Black Parrot Bar. This should be the largest rum selection available in London… that’s it - amazing place if you enjoy rum. Just note: the place is very tiny. You should make a reservation any day of the week, if you want to go - which I highly recommend.
Some general points, which might be helpful:
- it is always worth to check for the rum offering in any bar. Foursquare ECSs for example are very widespread and available at a lot of “good” bars and even restaurants
- places I mentioned above, are all also (and predominantly) cocktail bars. They will be very crowded on the weekends. So your best shot - if you are looking for a good conversation with bar staff and time to enjoy their whole range - should be to visit on workdays
- if you order a dram you will get c. 5cl by default. However, all places do offer half pours as well, which comes down to c. 2.5cl - just ask for those
- it is always a good idea to check pricing before you buy. Ok, this is obvious!
But my point comes down to “market differences”. Some bottles are much more expensive in the UK while others are surprisingly cheap. For example: The Appleton Hearts series is blisteringly expensive, while Foursquare is very accessable for most bottlings. So depending on where you are from, it makes sense to taste some bottles at home rather than in London
Hope this helps! Enjoy your trip and save some time in your schedule to engage in some bar seeing