You will need a stockbroker who is a member of the London Stock Exchange to carry out the transaction for you when you buy or sell shares. A broker is a go-between, in this case between you and the stock market.
Some large banks offer a dealing service to their customers so you can ask the bank where you have your current or savings account what facilities, if any, are available. The advantage is that you probably wander into your local branch from time to time to draw out cash or pay bills so you are likely to feel comfortable making an enquiry there.
You can buy with cash directly out of your account so there is no need to set up a new one.
The big drawback is that you may have to deal at best price, that is instruct ...