They do exist where there is commercial need for them.
First class capacity is already becoming a fairly finite resource on some operators, and the fact is that they don't need Advance fares to fill those seats / meet the needs of the market.
In this case, you essentially need GWR and EMR to reach a mutually acceptable commercial agreement on how revenue would be apportioned and how many seats can be made available at a certain fare point / tier - given both have different parent owning groups - it is likely that the revenue both parties would want from a first class ticket ultimately is around the price of the flexible fares that already exist or such a small number of seats would be made available, that the fares aren't really useable.
It is possible to get GWR and EMR first class itineraries to Luton/ Bedford.