I'm quite doubtful whether batteries alone can get a train up to 125mph but happy to be corrected. I think the aim should be to have most of the core routes electrified. I think Bristol to Edinburgh and Bournemouth to Manchester should be fully electrified. Ideally electrify through to Plymouth or even Penzance and eliminate the need for diesels or batteries. Maybe add third rail for Bournemouth workings?
A battery car can quite easily crack 125mph, why do you thing a train would struggle?
Battery trains will be able to comprehensively out perform an EMU as they are not limited in peak power by the OHL. A 5 car train 8xx style with a battery mass no greater than currently used for the diesels in a bi-mode could easily accommodate a battery capable of putting out 30,000bhp, the cost of putting a pair of performance EV motors on each axle is trivial as is the weight. The battery for a 5 car train would be about £400k so the economics are highly unlikely to favour electrifying new sections of track, you could put batteries in a whole fleet of trains for the cost of a few KM of electrified track.
I modelled the train going from Penzance to Leeds using IET energy usage on the ECML, it would need to charge from a short section of OHL while stationary at Exeter and Plymouth and would arrive at Bristol at 30% SoC. It gets down to 25% SoC before it hits the electrified tracks south of Brum and gets to Leeds at 24% SoC (unless I put some power in at Derby in which case it arrives at 37%).
My model is a little old and uses a Tesla Model 3 SR as the charge speed model, some of the newer Chinese EVs are now charging at rates more than twice that now so OHLE permitting it would take only a brief stretch of OHLE to top the battery off. This is a key point, EV technology is progressing at such a rate that something marginal today gets easy well within a planning cycle.
EV technology is definitely an example of a technology push, (where the technology occurs rather than being demanded from the customer) we can now make a train perform in a manner that no rolling stock designer, operator or network operator has previously considered feasible. It is quite possible to make a train accelerate at the limit of passenger comfort (1.3m/s) until it is doing 125mph+ meaning it can reach 125mph in only 1200m and 186mph in 3000m. It's quite possible a stopping service could outpace todays express services.
To really make these things work you need to vertically integrate, a rolling stock manufacturer can't innovate on tiny internal R&D budgets against rigid commercial contracts produced years ahead of time by an operator who doesn't understand the product, has no vision and limited control over the infrastructure. GBR need to get back into the train building business or at very least the train designing business.