Well as Mrs SDA tells me when she sets off on one of her shopping trips 'It's 20 minutes running time from Crewe to Chester'. As the Bear says you'd not get up to HS2 running speed before having to apply the brakes.
Series N700 Shinkansen are supposed to reach 270kph in under three minutes of acceleration.... but they are rather fast accelerating even for high speed trainsets.
I don't know Chester Station, but has it room for 400m long double decker trains that are wider than normal? Current platforms certainly couldn't cope. So as well as rebuilding the line and all its bridges you'd probably need a new station. And then the line couldn't be used by 'normal' trains so they'd have to go the long way round (Shrewsbury or Warrington maybe) or you'd have to build four tracks, two 'normal' and two high speed.
There seems to be a rather large number of platforms at Chester... you would only need two for captive high speed trains, and there are several platforms that could be stretched to 400m if you chose.
There is also the railway land north of teh station that could fit the required two platforms on (unfortunately it would be impossible to align them to allow for future expansions of the loading gauge further west without building a curved platform which I believe is now verboten.... but in the short term to allow captives to Chester it would do).
The width of the trains themselves is not a great deal of different with what we have here.... the Eurostar has 2.81m width trailers and teh TGV Duplex has 2.9m width trailers.
While the different platform heights would require seperate platforms it is unlikely the horizontal structure gauge woudl need significantly adjusting, especially as in this case there are no intermediate stations between Chester and Crewe as far as I am able to determine using Google Earth.
This means that you can rebuild the line itself to the larger structure gauge and then assign High Speed captive trains an appropriate routing prefix... (9Xxx?) so that Automatic Route setting can't mistakenly route Captive trains into the non Captive platforms.
Of course you could run Compatibles at high speed into Chester and that would allow the existing station to be used and you wouldn't need such a big loading gauge and maybe ordinary trains could run on it as well. Working that way you'd save, maybe, 4 or 5 of the 20 minutes. But why not just upgrade the line to modern Classic standards and be happy that Compatibles could go there anyway and probably quicker than they do now if they have up to date track and signalling.
As to costs I've no idea.
Because compatibles waste space on the more intensively used parts of the network. 400m Double decker trains are the future for low ticket prices and the like.
While you won't be able to run much above 110mph-125mph due to the curvature of the line you gain the ability to run full length trainsets into the station.
The cost of the conversion will depend on whether a several month shutdown to do all the reconstruction work is acceptable or if it has to be done in stop-starts.
In central chester it is likely you would have to dig out the cutting it is in to be 1-2m deeper while outside central chester it is more likely that bridge reconstructions to greater heights would be feasible.
If massive bridge reconstruction/digouts is required anyway it might not be massively more expensive than electrification would be.
EDIT:
I have now found a Network Rail document about curved platforms.
They can be permitted by waiver if "Platforms on horizontal curves with radii of less than 1000m may be permitted if due consideration has been made of stepping distances and visibility issues". Since ~300+m of the platform would be straight I think getting disabled passengers aboard would be accomodated on the straight portion of the platform no matter the stepping distances on the curved portion, however provision of equipment on the platform to bridge the gap at a door to allow access to a wheelchair space could be provided. Dispatch could be handed through appropriate video equipment.
So yeah, you can have a through pair of captive platforms.