I'm not a railway expert, or an electrical engineer, and no offence but if I was the government I would probably want 1. minimum spend 2. minimum fuss 3. minimum risk 4. non-speculative tech. So I feel like a few of the ideas above are a bit too wacky for them.
The GWR and Hitachi battery train trials have both demonstrated battery range of around 100 miles. As far as I'm aware Marylebone to Aylesbury is only 39 miles and Birmingham is 99 miles. GWR have been using rapid charging in around 4 mins. So not hard to imagine charge at Marylebone, charge at Aylesbury, charge at Birmingham. Can also do High Wycombe, Oxford, West Ruislip even if needed. Birmingham trains could stop for a few mins to charge at a station such as High Wycombe or Warwick, I've always found the Chiltern schedule to be rather well padded as it is. The infrastructure required would not need huge cost and closures for months or years, perhaps except a cost for the power network investment and charging in the station.
I'm sure it's a lot more complicated than that, but the idea of fitting rails like the London Underground, or overhead wiring at tens of millions of pounds each mile really could make it very unpalatable to the government. If Chiltern want this done before the diesel trains fully pack up, then I think it has to be batteries + fast chargers.
Am I confused or do Chiltern already have a tender out for the new trains?