I think it will take quite a lot of people fitting batteries and using mostly off-peak electricity to make a dent in the big disparity between the daytime demand (largely driven by industry and businesses) and the night time demand. The day/night demand variation is roughly 10GW to 15GW each day, as shown for the grid demand data for this month, and that for 14th November (gives a better idea of the times):

- November grid demand.jpg (96.68 KiB) Viewed 3703 times

- 14th November grid demand.jpg (81.33 KiB) Viewed 3692 times
If a million UK homes were to install batteries, and were each to charge those batteries overnight at 3kW for 4 hours, then that would increase grid demand overnight by about 3GW for that 4 hour period, and reduce daytime demand by perhaps 0.6GW for the remaining 20 hours.
I think we're several years away from home batteries having any significant impact on the balance of demand between day and night, TBH. EVs are most probably going to have a much bigger impact, and more rapidly, given how sales are going.
25 off 250W Perlight solar panels, installed 2014, with a 6kW PowerOne inverter, about 6,000kWh/year generated
6 off Pylontech US3000C batteries, with a Sofar ME3000SP inverter