It's not too bad here during the day. There are toilets in both museums, the library, a park and in Tesco and Homebase (althought the last two are right on the edge of town, not in the centre), plus 3 public toilets run by the council.
The museums shut quite early, the library is only open about 30 hours pw, and the park is only open until 5 in winter, sunset in summer.
The council-run toilets are locked at 6 or 7pm, so no chance of a piss anywhere in the evening unless you find a pub or cafe that doesn't mind non-customers using their lavs.
It's a 40 minute walk back to my house from the town centre, and in my drinking days I could never manage that without a pee on the way home. Luckily, we have lots of discreet alleyways and county hall car park is quite secluded with some mature trees to squat behind.
I had to have a pee on the M25 once, literally on the carriageway.
A lorry had caught fire and the traffic ground to a halt and I was stuck there for around 4 hours. And it wasn't one of the sections where there are bushes and stuff beyond the barrier, so all I could do was open teh front and rear doors of the car for a semblance of privacy, and squat between them.
Thankfully, I was wearing a dress, so didn't have to bare my arse to the anti-clockwise traffic. That sight could have a caused a massive pile-up.