I can only speak for my council, which complains it has no money but has hundreds of millions in slush fund "for a rainy day" and keeps building new council offices or rental spaces that aren't even being used or getting the rent. There's a huge one they built that needs like 30k+ a month in rent, no takers for 18+ months now (as you'd expect in such a crappy area and a certain pandemic).

They also have terrible web based services, akin to Frontpage '97 on a good day.

I'm not so sure, I think there is a real NIH syndrome in local government IT services, they quite often want to build something themselves - but then as you said don't have the budget to do it properly. Using the design system, and sharing platforms using it, would be a significant cost saving.

Plus the (at times) antagonistic relationship between Whitehall and local government I think means they don't want to necessarily use something they associate with the incumbent administration.

Not really.

Local gov doesn't have anything like the budget of central so they don't typically have the choice to build their own. Instead they're left with cobbling together behaviours from tens, maybe even hundreds, of disparate third-party systems, each independently solving the full stack of a given department's function. E.g. there's a dedicated system for planning approvals, another for administering blue badges, etc., etc.

These third-party systems are typically very cheaply made, with a very low maintenance budget, and don't provide anything like enough customisation to adhere to someone else's design system. If you're lucky you can add your own header/footer and set some main css colours to match your council's brand.