It is possible that passengers from Chesterfield and some from Sheffield will find it more convenient to stick with the 'classic' cross-country service. However for all points North - Leeds, York, Newcastle - HS2 will be an hour faster and will take virtually all current traffic.
The question then is, what will the 'classic' service look like? A half-hourly service from Birmingham to Sheffield seems possible, but would be designed to provide regional connections, so all trains stop at Tamworth and Burton. North of Sheffield, it would be useful to provide an improved frequency to Doncaster and then perhaps on to Hull/Grimsby to provide direct connections to Birmingham.