Because, when the UK government hands it out to the private sector, it gets the money back. All of it. Except, along the way, that money gets exchanged in lots and lots of transactions which the government skims parts off as VAT, Corporation Tax, Income Tax, NI contributions, various duties, plus a million other levies.
If the government "saved" money by choosing efficient suppliers with smaller headcounts and tighter cost controls it would cut off millions from the treasury coffers. Taxes which are desperately needed to cover the UK government's rising interest bill (debt is something like 95% of GDP as of 2025).
Huge behemoths like Fujitsu and Capgemini and IBM actually help to drive the UK economy in its ever more desperate drive for "growth" (i.e. greater tax revenue) and we can expect more, not less, wonga to be unloaded on them to provide crude "value" from which those precious taxes can be distilled back out.