Prior to OP's update of the question ...
Do I have a case to fight this in some way?
Without knowing whether UK legislation has a superseding provision in this regard, you seemingly have a meritorious position from the standpoint of contract law.
Your employment provided medical insurance as a benefit, [I assume] you accrued that benefit, and sought to use it while you were still employed there. To the extent that the delay in enforcing that benefit is not attributable to you (but to others such as the employer, the insurer, or the physician), the bills resulting strictly from that appointment are to be covered by whoever would be responsible therefor had the appointment occurred shortly after you requested it.
A contrary conclusion would imply that an employee's entitlement to accrued benefits is ultimately contingent on someone else's acts or omissions. That implication would generally encourage negligence --or perhaps deliberate obstruction-- in the performance of contracts.
The accrual & entitlement argument would fail if this sort of contingency were reasonably outlined in your contract(s) with the employer, insurer, or service provider. That being said, the form you were asked to sign during the appointment does not appear to inform you of that contingency.
You did not specify what the ombudsman responded. That information could shed more light on the legal merits of your position.
Upon OP's update/supplement of information ...
Do I have a case to fight this in some way?
No. The terms of the policy as outlined by the ombudsman are decisive (this is in line with one of my comments). I'm assuming that the ombudsman reading of the policy is accurate.
Moreover, the form you signed at the appointment reinforces the hospital's position that you are responsible for the expenses ensuing from the appointment. Signing that form placed on you the burden of first ascertaining an estimate of the expenses, it being the case that you knew --or should have known-- that you were no longer covered by policy.
These two last reasons --whether taken together or separately-- are likely to outweigh the merits of arguing that your appointment was delayed by having it scheduled for months later (see the initial version of this answer).
Proving that the bill is egregiously high in comparison with similar scenarios might be the only argument that could lead to a reduction of the bill, although this seems to be a stretch. Other than that, there is nothing you can do to reasonably dispute the bill(s) and surcharges.
That being said, I am keeping the initial version of my answer [that is, from prior to your update] because that rationale will guide others on how to assess akin scenarios where the issue is not determined by the terms of the policy or contract. Also, this showcases how leaving out one piece of crucial information can lead to the opposite conclusion.
