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We've got a built in wardrobe in one of our bedrooms which is situated next to a built in airing cupboard.

As the external walls are not insulated, the walls tend to get very cold. I noticed last night that in the corner of the built in wardrobe there is some mould starting on the wall and on some of the clothes.

If I was to fit some vents between the airing cupboard and the wardrobe, would the extra warm air from the airing cupboard help keep the mould/damp at bay?

only if you introduce some other vents, maybe into the door of the wardrobe so you get a 'flow' of air...
fitting a louvred wardrobe door would be an idea
or possibly 2 vents, one right at the top and one right at the bottom keeping the flow between them clear of any obstruction... this would allow warm air to flow into the wardrobe at the top and the colder air to be drawn into the airing cupboard creating a loop....
though i'd be careful of introducing damp items into the airing cupboard where the moisture would migrate into the colder wardrobe possibly worsening the problem.. dont forget warm air holds more moisture than cold air....
sticking some polystrene sheet to the wall in the back of the wardrobe would help...

I take it I'd need something like this?

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How is the sheeting stuck to the walls? Would I be able to use wallpaper paste? Can I just paper over the sheeting once it's in place?

I'll probably look into some vents for the doors as well.

gripfill would do better i reckon mate... just 3 or 4 long beads down the back...
or silicon - lot cheaper...
if theres any risk of fire stick a sheet of 9.5mm plasterboard over the top using same method....
then wallpaper or paint...

sheet of 9.5mm plasterboard is around 6 quid...

use acrylic decorators caulk round any rough edges....

better still just use insulated plasterboard.... its just polystyrene and plasterboard in one....

b&q might have a sheet or 3 in stock, if not try jewsons or travis perkins...

for some its a stock item, others dont bother and you have to order it...

if not just buy em separately and make your own insulated plasterboard with the old silicon..

p.s. if you do use silicon try and get a tube of 'low modulus neutral cure' silicon... it sticks like sh't to a blanket as opposed to the usual cheaper 'high modulus acetoxy' silicon that smells like vinegar...

for all i know wallpaper paste would stick it but ive never tried...