I am not single now but before dh I lived alone for years and years. I hated sharing with people other than someone I was in a relationship with. I somehow always managed to fill weekends and was busy all the time. Not always with other people but learning to be self reliant / happy in your own company is a great life skill.
In your position, being in London and foot loose and fancy free at weekends, I would make a couple of weekend a month 'adventures' - if you have a travel card you can 'be a tourist' in your own city!
Pick an area you don't know at all and spend the week researching things to see / do there. For me that would be a gallery or museum or building of interest. I would then plan my day around it by researching the best place for coffee in that area, a nice place to eat lunch or a park for a walk or a picnic, weather depending. A glass of beer / wine after seeing the museum etc and then home to some nice food.
You have SO many things to see and do in London. Do a history walking tour, I did the Tower of London one and it was excellent, really fun.
Do a life drawing class, or bread baking etc
Spend a morning browsing in a luxury shop like Liberty, ask for sample of the perfume, do lots of research spending as much time as you like in the jewellery department (can you tell that this is my dream!) Have a fancy coffee in there and people watch.
Go to the V&A and see the kiminos - I'd also love that!
Be brave and invite some people you don't know all that well to lunch / dinner and practice making something nice - it's a good way to break the ice and move a friendship to another level. I did lots of this when I lived alone. I invited friends who didn't know each other to a casual dinner on a saturday night and then that made it easier to suggest meeting for a drink or cinema or theatre on other times.
I think it really will come down to you having to make the changes