If you are travelling from Chinley station with your bike on either the Northern network or on EMR services, you can easily link up with other city regions - but will you encounter difficulties if you have your bike with you? And can you get to Chinley station by bus + bike?
The answer seems to be that folding bikes are welcome everywhere, but on checking the links given below, one transport operator gave rather vague advice which could cause difficulties for a traveller, and a number of bus companies state that they prefer folding bikes to be carried in storage bags.
High Peak Buses (owned by Centre Bus) operates services throughout the High Peak region and connects Buxton, Chapel en le Frith and Whaley Bridge with Chinley railway station via the 190 service. The company makes the proviso that folding bikes may be carried as long as there is space (info here).
Bus operator Hulleys of Baslow (Northern trains connect to their bus routes at Hathersage or Sheffield) sets out its rules here but on further telephone enquiry stated that it requires folding bikes to be carried in a storage bag.
In the Ashbourne/Staffordshire Moorlands area, the Moorlands Connect facility offers a bookable minibus service which can accommodate 3 bikes in a defined area of Derbys/Staffs.
Here is what Transport for Greater Manchester's Bee Network says about travelling with a bike. More info here...
Sheffield's Supertram also only allows folding bikes. See rule 29
... as does Nottingham's NET
Like Hulleys of Baslow, bus operators in Liverpool state that they prefer folding bikes to be in storage bags. Read more here for Arriva and here for Stagecoach.
On Merseyrail's new trains there are however designated facilities for all sizes of bike :)