Prescription drug prices in the US market are much higher than the NHS negotiated prices; without the US market and the high amount of US consumer spending on drugs, drug companies would have little incentive to invest in new drugs.
Your own doctors' and hospitals' inability to negotiate a good deal isn't the NHS's fault. You don't really think that if the NHS paid more the drug companies would say "oh well, we'll charge everyone else less", do you? They'll charge as much as they can get away with, just like now.
A more likely reason for drugs being so expensive in the USA is that they spend more on sales & marketing than R&D. How much cheaper would it be if they didn't do that? Don't blame your own dysfunctional system on the NHS "not paying up" because they do, and the companies make a fat profit out of it.