I'll have to check the details, but I'm pretty positive the USA has some protectionist laws on pharma pricing, it goes along the lines of it being impossible to get FDA approval unless the pricing is similar to existing drugs on the market for the same condition. Effectively pegging the prices artificially high.
As to US access to supply the NHS, that already exists and has for many years, but the NHS has some value criteria to obey in their procurement rules, while the prices are pegged high the suppliers don't get the contracts.
High drug prices in the US mainly have to do with the market there being actively rigged. Doctors get under the table payments from drug salesmen to prescribe certain expensive brands, drug store operators get payments from drug suppliers to put certain brands on the shelf and not stock cheaper alternatives, health care providers are forbidden by law from negotiating discounts collectively, etc. The buyers are fragmented and have no buying power, while a few big sellers dictate pretty much any price they want.
Americans like to think they have first class health care, while what they actually have is third world levels of health care corruption.
This "access to the UK health market" phrase is code for changing the laws in the UK to forbid the NHS from negotiating drug prices. Instead the American drug companies want to negotiate with each individual hospital so they can gang up on them to boost prices like they do in the US, bribe doctors to push certain brands, pay off drugs stores to lock discount brands out of the market, etc. The NHS's role would be limited to paying whatever the drug companies wanted to charge.
The drug companies tried the same thing during US trade negotiations with Canada, so I've heard it all before. The US got told to ram it, and other US economic interests didn't want to give up a trade deal just so a handful of drug companies wanted to line their own pockets, so it got dropped.
The political system in the US is corrupt and revolves around money and whose pockets get lined by which corporation. If you think that you can buy off one of them and that will be the end of it, then you'll be in for a shock. The rest of them will pile on you like a school of piranhas looking for their share as well. It's like paying off mobsters. They'll keep coming back and wanting more.