London is actually one of the easiest places to make new friends because the city has a large transient community and because of this people are always looking to develop new friendships.
So while the fast pace of London gives off an impression of a cold city, don't worry part of the fun of London is the new friendships you will make.
Play a Sport, join a sports team/Club.....If you play a team sport this is an easy way of making instant friends
Speed Flat-mating/Live in a house share......House share is not often only a cheaper rental option but its also a way of finding new friends both with the house mates and associates you meet through them. Speed flat mating is a way of connecting with people before you decide to live together and can also be a source of friendship even if you don't.
Dine at a Supper Club...Many supper clubs are communal events where you get to experience great food but also meet new people whom you are seated next to.
Look for events within your expat community (or another)....Attending expat events in London is a great way of meeting people, even if they aren't from your own expat community as there will still be lots of people in the same boat as you.
Take up a hobby....This is a great way to meet new people and they are likely to have similar interests.
Join a Fitness Club....The gym can be an ok way to make new friends but being involved in regular fitness classes such as Zumba or British Military Fitness. Don't make friends at least you will be fit.
Become a local at your favourite pub.....This can be a great way of meeting people and developing friendships.
Get a dog.....This is impractical particularly if you are an expat or renting but having a dog provided its not an aggressive beast is a great way of being introduced to new people.
Go on a tour.....This could include a pub crawl such as a Monopoly pub crawl or multi day tour somewhere exploring parts of the UK or Europe. Either way you will have a new set of acquaintances that you will be sharing an experience with.
Attend Networking events...This could be a business, educational, charity, expat or hobby event either way people are there looking to make introducations and develop relationships.
Find friends online.....Dating websites and website forums can be a way of making new friendships.
Stay in a Hostel....Hostels are full of travelling backpackers who are always sociable and many hostels run events to bring the short term residents together.
There are lots more things you can do make new friends, give yourself time to build a good base of friends in London and don't be discouraged when some leave London as new ones will come along shortly.
********* Remember most people who live in London are not from London, there is a big chance they are looking to make new friends just like you. So if you meet someone whose company you enjoy don't be afraid to ask them to hang out with you.