I recall GDS being initiated for internal DOH stuff circa John Major, going into Blair era (yip was there when Blair took over the role).
Not that I'd attribute it to any PM, believe it was driven by the government security services (which department/flavor I could not say).
That all said, it may of been a different aspect but certainly has a longer story than told. With that, it started as an intranet project iirc.
https://gds.blog.gov.uk/story/ has some insight but that seems more to cover the more public web aspects and even they say it is "a story" not "the story".
So be fascinating to peace the "full story" and history together.
GDS was a whole catchment in the early days, that included the phone services and that was contracted to Mercury telecom at one stage, or aspects were. https://en.wikipedia.org/wiki/Mercury_Communications
If any credit should be laid, it would be upon the civil servants and associates that came up with the initiative, who sadly will probably remain nameless even if their names were known. Politicians just green light so much and claim credit, so much goes on behind the scenes of any government. With that, best think of politicians as orchestra conductors who depend upon musicians knowing how to do things, often able to do things right without eye sight of the conductors at times.