I think by the point you are ready to hire a big foreign consultancy you’ve generally tried in-house and failed.
Of course you can try with a local consultancy, but I wouldn’t know of any, and I assume the reason for choosing one over the other is mostly a matter of reputation.
I think you’re giving the British government far too much credit. Between 2010-2024, the governing party had a religious belief in the power of outsourcing to the point that former ministers (Rory Stewart being one) have spoken out about how many problems they caused by pushing the model to its extremes, irrespective of whether it makes sense or not.
Where it comes to organisational complexity and the barriers it creates, bear in mind that the British state is vastly more centralised than the USA. Fragmentation in the NHS was massively exacerbated by the Lansley Reforms which also forced trusts to outsource a lot of work.
Of course you can try with a local consultancy, but I wouldn’t know of any, and I assume the reason for choosing one over the other is mostly a matter of reputation.