I have a one-bedroom flat in which I want to install a £1 coin meter
for the elctric supply. Can these typical £1-coin meters safely take
the load of an electric shower plus an electric cooker (plus the usual
lighting and ring main)?

They should usually state the load rating near the serial number of
the meter.

Alternatively, does it have a "red spot" on there somewhere ? (often
around the glass) - IIRC that means they were uprated from 30A or 40A
to 60A (might even have been 80A) back in the 80's.