The 47 GW peak which would provide more than enough power for 30 million cars was this year. I gave the all-time peak as an example of how high it has been in the past - as a means of demonstrating that not only is charging cars within the scope of our current grid, but our current grid isn't even as heavily utilised as it was in the past - so it's unlikely to be burdensome to expand it back to those levels.

I don't know to what extent the infrastructure itself has changed - but given National Grid themselves don't think it'll be a big issue, the grid itself is probably still capable of that 62GW (even if generating capacity is less).

I haven't mentioned CO2 in my comments because that was not part of the original discussion. "Its not even clear how we are supposed to produce the necessary amounts of electricity" is patently false in the case of the UK. Maybe overall CO2 production would be less, maybe more - that's a different (though important) discussion. At least with electric vehicles there's a clear technological roadmap to 0 emissions - there's no credible roadmap for 0 emissions with internal combustion engines.

Vans aren't likely to be vastly different from cars - eg the Ducato Electric claims 235 km on a 47 kWh battery - vs a Model 3's claimed 265 km on a 50 kWh battery. HGVs are much heavier vehicles though - the path for electrification for those is likely to be slower.