You said the BSGE centres are all NHS hospitals - that is different from saying the majority are. The BSGE is not connected to the NHS, it is independent. There are more private centres evolving as can be seen from the provisional list.
Any excision surgeons operating on severe disease on the NHS as defined by the NHS England service contract (E10/S/a) who are not working out of centres applying the BSGE criteria are breaking the terms of both the contract and the 2016 RCOG Standards for Gynaecology Care, which they are bound by their RCOG registration to apply.
With regard to private care, surgeons must apply the same standards as laid out in national guidelines, as would apply in the NHS. The CQC confirms that in assessing independent hospitals the following must be applied:
1. How are relevant and current evidence-based guidance, standards, best practice and legislation identified and used to develop how services, care and treatment are delivered? (This includes from NICE and other expert and professional bodies).
2. Do people have their needs assessed and their care planned and delivered in line with evidence-based, guidance, standards and best practice? How is this monitored to ensure compliance?
Clearly we would not be happy to be treated by private surgeons failing to apply evidence-based standards?