In reply to AndrewInVancouver:
I've ridden parts of it around Liverpool, Manchester and Sheffield on a Brompton with no problem. However the TPT website says this:
Most of the Trail is off-road, but not all is surfaced; you will need a bike and tyres capable of being taken off-road. A mountain bike or sturdy tourer will be more than capable of riding the Trail, thin racing wheels / tyres are not recommended.
You could try asking the TPT office for more information about the unsurfaced areas. You may be able to find an alternative route past these.