The Chester will still be, in effect, a slow service with just not as many stops as current. It certainly isn't becoming a fast express service and hence that's why Sprinters will be adequate. End of the day if you want a fast service to Stoke or Chester then you take a Virgin/Cross Country to Stoke and a Llandudno service to Chester. It's more the case with Chester because of the low speed of the CLC route, with Warrington route much faster.
It's baffling that ATW want to make the Stoke service faster, when passengers already have the choice of 4 express trains per hour to Stoke and 2 to Macclesfield. Northern should focus on an increased service frequency for the intermediate stations en route, not skipping stops. Same applies to Chester via CLC; apart from perhaps Ashley and Plumley, both services (Greenbank and Chester) should be all stops.
You misunderstand the aim of speeding up services. It's not to get Stoke to Manchester passengers on to Northern services!! There's two things to consider:
1. Faster journeys from places such as Congleton to Manchester. In the case of the Mid-Cheshire line the service does carry a number of people from Chester to Stockport, from where they make connections to other services mainly the Sheffield bound services, which is one reason why a direct Chester to Sheffield service was considered. While speeded up services will allow Arriva to market places such as Buxton and Delamere Forest as being under 1 hour from Manchester by train.
2. Having the new Macclesfield to Manchester service as a semi-fast and keeping the Stoke to Manchester as all stops would likely mean the loadings would be very unbalanced, so reducing calls on the Stoke service and running the Macclesfield service as all stops should help balance out loadings.
The ITT included requirements for extra calls at stations like Mobberley at rush hour and at school/college finishing times. I don't know why you think they need 2tph all day, when some off-peak services stop there without picking up or dropping off a single passenger, yet the same service can get 40 passengers using Knutsford. It doesn't make sense for Mobberley to have the same level of service as Knutsford. Mobberley station isn't even well located to serve the village, the 88 bus route between Wilmslow and Knutsford serves the village better and the bus service is half-hourly and usually operated by 64 reg vehicles with a high specification interior.
With you saying 4 car Sprinters to Chester and Buxton it didn't sound like you realised the ITT requirement for additional off-peak services on both lines. While I hope the more frequent services attract more passengers I don't think quadrupling of capacity is required yet! Buxton will still need some 4 car workings (ideally 4 car 156s) to Manchester in the morning peak and away from Manchester in evening peak, but during the Mon-Fri off-peak period 4 car Sprinters every half-hour to New Mills Newton, on top of the Hazel Grove only services would be overkill.
The Mid-Cheshire will be a bit more complex. On a weekday a mix of 2 and 3 car services together with the service enhancements should provide enough capacity. However, with Chester only getting 1tph on Saturday if there's races on in Chester a 3 car train would be too small, on race days a 4 car 150 can have standing passengers currently.
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Just crossed my mind - might the 333s be refurbished with 2+2, tables etc and used on the electric Northern Connect services? All the new EMUs could then be to a higher-density layout for suburban journeys both sides of the Pennines. Are 333s due a refurbishment?
The 333s will need some kind of refurbishment as they have to meet the latest accessibility requirements and have an interior which has a 'like new' look and feel.
However, I think due to PTE agreements they have to remain serving West Yorkshire until at least mid-2020 and Northern Connect standard trains have to be in place by 1st Jan 2020.