A simple solution to overcrowding on Sheffield etc expresses is not to advertise the stops and deny boarding during peak hours on East Mids trains at St Pancras and to stop no trains southbound during the morning peak.
This is what happens on WCML. It allows Virgin and London Midland to compete and lowers fares at Milton Keynes during the off-peak periods. Couldn't the same be done at Luton and Bedford?