Or there's always the paid Canonical support. But the thought of David Cameron hunched over a laptop having some tea and getting assistance from lazer420 in saving the NHS from the next heartbleed is priceless.
No, they ask their IT department.
The difference being, they can fix issues with the OS themselves (or hire someone to). With something closed like Windows, there's only one place to go for support.
Access to the source code is one thing, but you'd need also developers who are familiar enough with the source to be able to actually make changes. With OSS, there is a large community from which you recruit these folks, with Windows, not so much (outside of MS).
It's not the license cost that's keeping these people from upgrading. It's verifying that the hundreds of applications, many of which were built by systems integrators that you have a strained relationship with that need to work correctly on whatever you're replacing it with. It's training users - some people "need" a two-day offsite seminar and a couple of weeks of reduced workload to transition to a new system. Also, something invariably goes wrong, so you also have to account for lost productivity when that happens.