Having been through a few of these public sector transformations as a consultant, there is no "there," from "here."
Strategically, the play for a few successive governments has been to stand up new arms length agencies outside the public service proper that have missions to develop new platforms, with the intent to migrate the public service to them when they arrive.
The successes have been where the stand-alone agencies manage to deliver and maintain their independence. The failures have been where they have become captured by "secondments," where mandarins or "retired" ones leverage their way into governance roles, subordinate them to the old system, and turn them into the same broken and staid institutions they were designed to replace.
The successes were expensive but their success amortizes that cost. The symptoms of the failed ones were visible early on, generally because of Party-connected consulting firms owning the HR contracts for them and using them as patronage factories and slush funds.
Change requires freedom, and that freedom requires independence. The accountability risk is real, and the challenge with oversight was that the overseers simply folded the agencies back into the morass.
If I were running this, I would take a radical approach, where the strategy would be a venture model that micro-funded projects to create useful products that solved for providing future services and applications, and not legacy ones.
Priority one is a set of useful and privacy preserving identity services with varying levels of assurance that meet the needs of regular people, not just public servants, as all public servants are citizens first, so are researchers, academics, police and everyone else. This must be exogenous and adopted willingly, and not forced down as a national identity scheme. Second would be an investment in FHE and other privacy technologies because they can preclude and even obviate the need to stand up new private infrastructure to support data exchange. Third, I would invest in federated and distributed techs (git/BC/OIDC, etc) because interoperability is the ONLY problem government is designed to solve well because fundamentally it is a coordination and interoperability layer for society in everything it does.
While I am not qualified for this job, with a long enough lever, I have no doubt I could move the whole thing.