Re: actually no
The easiest fix is already in place. Electricity providers mostly offer reduced rates to EV owners if they charge their cars off-peak in the wee hours. Every modern EV I've seen can be set to start charging at a specified time. Some have a feature to default to the programmed time when at home and to start immediately anywhere else.
A 45 minute commute each way for work is likely only 30 miles or so one way. There should still be plenty of range for an evening's toodle around the shops or cinema or the person bought the wrong car (ie, a Fiat 500e).
It's best to thing about charging in terms of how many miles of range are replaced for every hour of charging using different power sources. In the US, with half the voltage, it starts at 5mi/hour for 120V, 25mi/hour for 240V and gobs for dc fast charging that will vary with its power output rating. The slowest rate in the UK is about 10mi/hour. That's got the car topped off from the 60 miles of use to commute in 6 hours. If your charging started at 10pm to take advantage of better rates, the car is all done covering the commute by 4am while you are still sawing logs. Nothing is mandated that you have to completely top up the battery every time you plug in to charge so if you wind up driving more one day, it's not a big deal. The more level 2 chargers that wind up in shopping centers and along the high street, the easier it will be to follow your ABC's (Alway Be Charging).
Robert Llewellyn has a great show on YouTube called "Fully Charged" and there is an episode from way back where he visits the control center for the national grid. When he asked them about EV charging, he was told that they'd love to have more usage in the middle of the night to fill in the "bathtub" of reduced demand. Keep in mind that refining crude into petrol takes 7.46kWh per US gallon of electricity. Just the electricity to make a gallon of petrol can push a Chevy Bolt EV a bit more than 30 miles. The electricity used to refine 8 gallons of petrol will charge a 60kWh from flat to full. The range of the Bolt is advertised as 240 miles (238 really but I'm rounding off). 60kWh @ .12p/kwh is £7.20. How much is 8 gallons (30L) of petrol to push an ICE car that gets 30mpg that same 240 miles?
I had the same thought about everybody plugging in and charger when they got home but learned that it's really a non-starter. People get home and put the kettle on without a second thought and that is very noticeable on the grid. You won't like the results of making your coffee or tea at 2am to get a better electrical tariff, but it isn't a problem when charging the car.