As per the title, does anyone know if lotus cortina wheels need inner tubes? Friend of mine has a set for his angling and they do not seem to have a bead.

When in doubt ,take it to a tyre shop and get advice...

Could well be that the rim does not have the seat for tubeless. have that issue with my Facel on Borrani steel wheels - ended up using a rally beading gum that glues the rim to the tyre. Not ideal, but depends on the tyre you want to use, some tubeless are too rough to take a tube

You should definitely fit inner tubes in Borrani wheels. 
 

even though some wire wheels claim to be tubeless I have seen too many failures to trust them. I suggest always putting inner tubes in any wire wheels. 
 

you should not fit an inner tube in low profile tyres. But that doesn’t matter because you shouldn’t fit anything like a 65% profile on a classic car any way it will not handle nicely. 
 

any full or 70% profile tyre can fit an inner tube regardless of weather it is described as tubeless or not. 
 

so any tyre that is suited to a Lotus Cortina can fit an inner tube if you want. 
 

my guess is that your wheels don’t have a safety hump so a tube is right. 
 

the film on here should help https://www.longstonetyres.co.uk/classic-tyre-inner-tubes.html

 

However you wouldn’t ever want to fit anything lower or wider than a 185/70-13 so that is fine.