Do the free one. If you don’t feel it covered what you wanted to cover either do it again or pay private.
In my experience NHS is best for emergencies, general checks and enquiries.
Private is best if there is a specific problem that needs dealing with.
Example : A ‘person’ fell and injured their shoulder with various other cuts and bruises. NHS x-rayed and saw no breakages, patched up the wounds and sent them home. After a week the shoulder was getting more painful, not less and went to GP. Told it was probably just bruising as the x-ray showed nothing. Wait 4 weeks and if it hasn’t improved then come back. 4 weeks and it was worse still so got a referral to physio and waited 6 weeks in pain. Physio said it was a damaged shoulder and not their area so sent back to GP. GP referral to another physio. Physio 2 said there was severe damage inside the shoulder that would require surgery. GP sent the person to a specialist to discuss surgery. Wait 8 weeks. Chase to be told it was delayed so wait another 4 weeks. Appointment cancelled so wait a further 2 weekx. Specialist says that because of the age of the injury the only option was to cut the tendon to relieve the pain but the person would never raise their arm above their shoulder again.
Went to the private specialist the following week who suggested building a support channel for the tendons, repairing them and then 3 months of physio. Surgery the week after that with immediate physio when required and shoulder fully rtepaired.