An analogy might help here:
Let's say that you call a plumber round to your house to fix a leaking tap and (unwisely) forget to ask him how much he'll charge for the work first. If he tries to charge you £10,000, you can refuse to pay on the grounds that the charge is not 'reasonable'. If he then tries to sue you for the money, a court will almost certainly strike out the claim, again on the grounds of 'reasonableness'.
Equally though, if the plumber tries to charge a much lower sum, you can't then simply say, "I never agreed to pay you a penny before you started work, so you're not getting anything from me now". In such a situation, a court would rule that the plumber was entitled to 'reasonable' payment for his work.
Under the Principles of European Contract Law (which have been broadly adopted by the UK, even post-Brexit, and which generally reflect English contract law anyway), "reasonableness is to be judged by what persons acting in good faith and in the same situation as the parties would consider to be reasonable. In particular, in assessing what is reasonable the nature and purpose of the contract, the circumstances of the case, and the usages and practices of the trades or professions involved should be taken into account".
What that essentially boils down to is that if you ask a private health practitioner to carry out a procedure, it is reasonable for him/her to expect you to pay a charge for it that is broadly in line with what similar health practitioners would charge for such a procedure. As with my plumber analogy, it's not necessary for the person doing the job to tell you in advance how much it will cost. The cost simply has to be 'reasonable' in all of the circumstances.
With blood tests, the situation is further complicated by the fact that doctors don't have their own pathology labs. They have to send a sample to an external company and pay the fee charged by that company. They then add on their own fee for taking the sample and sending it off.
The fees charged by labs can vary widely, depending upon the nature of the blood test. For example, Nuffield Health will only charge you £32 for an 'online' blood glucose test but a 'comprehensive liver investigation' blood test costs £334:
https://www.nuffieldhealth.com/pathology-direct
So, without knowing what the doctor asked the lab to test for (and what the lab then charged him for carrying out that test), it's impossible to say whether or not the amount you've been charged is 'reasonable' or not. If, say, the test was only for blood glucose, a lab fee of up to £60 or so, plus perhaps another £50 to the doctor for taking the test and sending it off, might be reasonable (meaning that you could possibly argue that you've been overcharged).
However if the blood test was a very complex one, they might reasonably charge perhaps £400 for their work, plus another £50 to the doctor, meaning that you've actually been considerably undercharged.
I suggest asking the health provider to break down the £250 fee for the blood test into its two main components. i.e. the amount paid to the lab for the test and the amount charged by the health provider for taking the blood sample and sending it off. If the latter part looks grossly excessive (such as, say, £200), you
might then be a better position to argue for a reduction but it could still prove difficult to argue such a case in a court of law. If the health provider states that such a fee is broadly similar to what other health care providers charge, it might still be regarded as 'reasonable'.