No. It is not NHS but a NHS-commissioned technology provider owned by Square Health a provider of private digital healthcare for insurance companies and corporates. In other words GP services being privatised by stealth.
In June 2017, the Care Quality Commission (CQC) found Push Doctor to be providing caring and responsive service, that fell short in areas of safety, effectiveness and leadership. The Commission suggested that Push Doctor were prescribing high-risk medicines to patients without performing essential checks, and in one case had prescribed a medication for a use beyond its stated license. Then-CEO Eren Ozagir claimed that there were factual inaccuracies in the CQC report, and that the guidelines against which the inspection took place were completely new.
In November 2017, the company failed a CQC inspection due to inadequate checks of users' ages and identities – which was then corrected in time for the August 2018 inspection, wherein the company was found to be overprescribing antibiotics.
In July 2018, it was reported that Nicola Blackwood, the former minister for public health and innovation, had joined Push Doctor's advisory board. On reviewing the former MP's new role, the Advisory Committee on Business Appointments stated there was some risk around the potential for Blackwood offering the company an unfair advantage as a result of her time in office.
More here: en.wikipedia.org/wiki/Push_Doctor