Countess of Chester Hospital NHS Foundation Trust
The trust website says that private fertility services  are available:
As an alternative, you may wish to consider private treatment, which the Countess of Chester can also offer. However it should be taken into consideration that private treatment for infertility can be very expensive, and as with NHS funded treatment, there is no guarantee that it will be successful.
The trust has several private, or potentially private, income streams. The Forward Plan  says;
Following the expansion of the Trust facilities we will pursue opportunities to extend the volume of private patients and opportunities to extend the volume of planned operations to commissioners.
The Forward Plan also says that the trust have partnered with a private company, Gravitas:
we have bid in partnership with a private sector partner to undertake the Bariatric surgery contract for NHS patients in Cheshire.
The surgery provided by Gravitas is not just for NHS patients, they say they will:
Create an additional ward facility of approximately 30 beds in the new build to accommodate the Bariatric Surgery Centre and allow expansion of private and NHS elective activity.
The trust also have partnership with an NHS provider with a joint venture called MicroPath and there is an implied suggestion of this company doing private work:
The Trust has created a shared service in Human Resources and a joint Microbiology laboratory, MicroPath, through its alliance with Wirral University Teaching Hospitals NHSFT which we aim to market more widely.