That depends, there are multiple considerations:

One is access for "everyone". Many don't need that.

The other is access to a large enough labour pool.

That said, the easiest way for government to use government departments for district development policies, is to simply move them.

E.g. this is where the Norwegian equivalent of Companies House plus assorted similar registries is located:

https://www.google.com/maps/place/Br%C3%B8nn%C3%B8ysundregis...

If you zoom out you'll notice there are no major cities nearby.The nearest major city - Trondheim - is ~6h-7h drive South. It's smaller than Nottingham.

Norway has "just" relocated government departments out of Oslo as an intentional policy for decades, and it takes effort but it generally works quite well.

Transport policy then follows depending on how this changes transport patterns, rather than lead.

That depends on what institution is moving. Some just need a good link to Manchester, and so will be more willing to move. Some need to get everywhere and will wait. It depends on what the pros and cons of each city are.

As to if enough will move given any particular link, that is a hard question requiring complex studies and so cannot be answered.

Yep, and a good line from Liverpool and Manchester to Leeds would be a very good start. As then you good access to Leeds from many of the highly populated areas of the UK.