Hello, I'm a UK citizen moving to the US and will be transferring to the US benefits scheme. Can someone explain to me how it works in terms of getting medicine for an existing condition (i.e. not one off visits to GP).

Going by your name, I guess you are coming in on an L1. Presumably your company offers health insurance. An amount of money will be taken from your paycheck pretax to pay for this.

From there it depends on the type of plan. Generally speaking you register with a primary care physician (GP) who is in your insurance network (a simple visit may just be a copay say $20 or a proportion of the cost until your deductible is met). You will go to see him/her tell them about your on going condition and ask for an Rx script. How you insurance pays for this will depend on what tier the drug is in, generic, brand name etc. (could be anything from a few bucks to full list price out of pocket).

I just want to add that depending on what it is, your primary care might not want to manage the ongoing condition and so you may have to find a specialist who will. Your primary will probably be able to give you referral if your plan requires it or a recommendation if it doesn't. Be sure to check that the specialist is also covered by your insurance

That is if the company offers a prescription plan on top of the medical plan. Most often it is two separate policies.

You also need to determine if the medical plan is an HMO or a PPO. In either case if I am reading your original post, you need to have a primary doctor and if you don't require a specialist, then the primary issues the prescription either by written slip but usually by calling your pharmacy and ordering the prescription for you. If it is a drug that you take every day and you need a renewed prescription, usually you can call the doctor's office and ask for a renewal to be called into the pharmacy. That is the way I handle my diabetic drugs.

I get my long term diabetic medications and supplies in the mail via CVS Caremart, doc sends prescription for 12 months and CVS ship me 90 day supply at a time with auto refill, refills arrive in time that I don't run out. The cost savings of 90 days versus 30 days at the local pharmacy are 30% or more plus I don't run out or have last minute panic run if I have a business trip. They have also encouraged me to ask my doc for generic options where possible to further reduce my costs.
Maybe your health benefits will be similar, you need to ask