What is a police force going to do with an IP address? As I understand it they would need a warrant to obtain the subscriber's details from an ISP, they can't just call up Vodafone or BT and get a name and address.
Plod can obtain the name associated with the account tied to the IP address at the time the words were posted via a warrant as you rightly surmise.
But here's the thing. Just because cp8759 pays the bill for his/her internet access, it doesn't mean he/she posted the bad words. He/she simply says, there are x people in my house using my internet, it wasn't me. And Plod have to go away with their tails between their legs.
Equally, and here's a real world example, I allow my internet to be accessed by my neighbour via wifi (I've given her the SSID and password) - she's on her own, works hard but has next to no money and can't afford her own internet access.
What she does with it is down to her (I've told her no video streaming during the day when I need my internet for work).....but other than that, if she posts on Twitter that she wants to nuke the UK Government and the trail comes back to me, I'll tell Plod to go forth and multiply, not my problem.
And all of that is before you get into VPN usage.