That's simply untrue - National Grid don't forsee a problem. [1] Peak power usage in the UK was in 2005 - energy saving measures have been reducing it ever since. [2] The grid is today only delivering around 70% of the power it delivered in 2005.
Electric cars individually may have high peak charging loads, but if you calculate the total average power draw per car over a years driving, it's only about 200W. We may have to implement some level of demand management either in the cars or car chargers, but that's not particularly complicated - all the infrastructure already exists.
I do agree that the charging infrastructure is a long way off. Every street of terraced houses needs to be dug up and wired up with regular spaced charging points.
[1]: https://www.nationalgrid.com/stories/journey-to-net-zero-sto... [2]: https://theconversation.com/britains-electricity-use-is-at-i...