If you decide to take a break from drinking tea every three minutes, London has a coffee shop on literally every corner. Much like Starbucks (which they also have here) is to New York City, Cafe Nero, Costa, Leon, Illy, Apostrophe and more are to London.
If you take the coffee away instead of drinking it in the store, it will be cheaper (there’s a tax that the food venues are required to pay that comes with eating in the restaurant. So, if you drink your coffee at a little table, even if you picked it up from the counter yourself, you still have to pay as if you were being served by a waiter) than staying in the shop, although the atmosphere is usually very pleasant inside.
Most shops will not advertise a black coffee option. If you don’t want a “fancy” drink (cappuccino, americano, latte, mocha, etc.) you’ll have to ask for “filter coffee” and ask for it to be filled to the top (they will still leave you enough room for milk despite your request). These are usually pretty cheap (around £1.10), but you need to know to ask for it even if you don’t see it on the menu. They almost always have it, but prefer to sell the more expensive drinks listed above (those can range from anywhere between £2-£4 per drink).
There are also ALWAYS 100000000 tea options if you prefer to drink like the British do (outside of a pub).