I'd immediately say with the arrival of HS2 that the Intercity service pattern needs to change or they will be competing too much. Having the East Midlands Cities with that faster London Connection means you can offer more capacity of London-Bedford at a faster speed through the Intercities.
HS2 isn't about the East Midlands - in fact in many respects it doesn't offer much in practical terms.
HS2 reckon London - Birmingham will be 49 minutes - currently the MML covers St Pancras to Leicester in 1h 03 mins. Nottingham - Leicester can be done in 20 mins currently, so you're looking at ~ 1h 20m London - Nottingham.
At best HS2 will match that - Nottingham - Birmingham is about 50 miles as the crow flies - you're still going to be looking at ~20/25 mins Birmingham - Nottingham HS2 plus reversal time at Birmingham.
Bedford is not a significant place - there isn't the demand to travel to it - there are no large employers based there. Similarly there is minimal northward commuting from Bedford, primarily because Wellingborough and Kettering don't have much in the way of large employers. Leicester does, but in the time you can get to Leicester, you can also get to London - and London salaries are *much better* than Leicester ones.
I had a post about this a while ago where I had worked out the fastest time from the East Midlands Cities to the destinations on East-West Rail. You look just at Leicester which is by far the one with the best existing connections to the EWR destinations due to being closer. While Sheffield, Nottingham and Derby (which have a combined population of around 1.2 million.
Even if you look at Nottingham or Derby to MK, there's no practical improvement going via Bedford and EWR.
Nottingham - Tamworth is 54 mins, Tamworth - MKC is 52 mins. Total of 1h 46m.
Nottingham - Bedford would be 1h 16m. MKC - Bedford via EWR will be circa 30 mins.
Derby's it's even more in favour of the WCML route.
Derby - Tamworth is ~ 20 mins, so you're looking at 1h 12m, compared to Derby - Bedford at ~ 1h 15m - basically by the time you get to Bedford from Derby, you could be at MK going via Tamworth.
You state there is no advantage of travelling via EWR to Cambridge as they can already connect with the ECML. Well taking the current timetable today the fastest journey is actually taking EMR to Ely and then changing to Great Northern or Greater Anglia. This has a journey time of 2 hours and 5 minutes. If I set this for arrival at Cambridge station at 11am from Nottingham the fastest route is 1 hour and 56 minutes taking that exact route of EMR to Ely and Greater Anglia to Cambridge.
As the service pattern proposed is for the 2tph Intercity trains I will take the stopping service:
https://www.realtimetrains.co.uk/service/gb-nr:G15176/2023-03-08/detailed as the base for working out timings. There is the addition of Wellingborough which I would expect to add 5 minutes to the journey and then 2 and a half minutes for the stop at Bedford to alight. So the time to Bedford would be 1 hour 16 minutes and 30 seconds and EWR state they want it to be 30 minutes from Bedford to Cambridge so add the even longer connection of 30 minutes it comes to 2 hours 16 minutes.
But the service from Nottingham to Ely is an hourly service so a desire to arrive at even slightly off give it to this route.
First up, the Eastern section of EWR is many years away - and I think their "30 minute Bedford Cambridge" aspiration is optimistic. The line's going to be in the region of 30 miles in length and will have at least 2 stops - Cambourne's a given because the Cambridgeshire authorities won't support it if it doesn't, and I suspect something in the vicinity of where it crosses the ECML will also happen, supported by a load of housebuilding in the area. Allow 5 minutes to cover those stops and they're looking for an *average* speed of 72 mph. For comparison, Stevenage - Cambridge with 4 stops is taking almost 40 minutes using 100 mph EMUs and using part of the route EWR's likely to need.
Given the ECML is likely to see improvements in speed and capacity, particularly given that HS2 isn't going to Leeds now, there are doubtless going to be opportunities to speed up the Nottingham - Ely service.
Currently Nottingham - Grantham takes 31 minutes (22 miles), so averaging about 40mph, Grantham - Peterborough takes 27 minutes (29 miles), so averaging just over 60mph and Peterborough - Ely takes 33 mins (30 miles) so averaging just under 60 miles. That journey is 81 miles and taking just over 90 minutes an average of 53 mph - if that could be improved to an average of 70 mph, that journey time would drop to 1h 10m - basically getting you to Ely in the same time as you would to Bedford from Nottingham. And improving those linespeeds between Nottingham and Grantham and Peterborough and Ely would offer far more benefits than slowing down long distance EMR services to stop at Bedford for the one or two people who want to change to EWR eastwards (assuming it ever gets built).
As ever when there is the suggestion of putting Bedford back into the long distance EMR services it's always a case of a solution looking for a problem - this is no different.
Putting the Luton Airport Parkway stop in, I might, just about, agree with. I think there may be some benefit to that. But not Bedford.