Generally speaking the NHS pay drug manufacturers the same price for their products as private clinics. There the similarity ends. The costs passed to you and I, the consumer, by private clinics will vary between individual clinic. In England of course there is a standard prescription charge unless you’re exempt.
Whilst I am will to go with the "generally speaking........", there are a number of exceptions. This now, but for some years Liothyronine/T3 (a thyroid hormone replacement) was £9 per TABLET to the NHS, but less than that, supplied by private prescription - for those who could afford it. (In some places overseas it as pennies per tablet.)
This led to T3 being withdrawn from NHS prescription, except in exceptional circumstances. This in turn led to many people, mainly women, being undermedicated. In my area, my Endo was near celebratory in telling me there were 11 patients under our Trust being prescribed T3, The context was because I was the 11th patient. For me, T3 is a game changer, in a positive way.
In recent times, T3 has been reclassified again (spookily enough shortly after a renegotiation of the contract - reducing the cost, a bit), but is still a consultant only drug and their purse strings have not been noticeably relaxed.
In many areas, the NHS procurement procedures are shambolic, and in this example, near criminal.