None of the images I can find show an expiry date on the card.
e.g.
http://www.nfsp.org.uk/write/MediaUploads/General%20news%20images/POCA_2008_022.jpg
Further, the Ts&Cs say this:
"If you suspect that your card is damaged, take your card to any Post Office branch and they will be able to replace it for you there and then"
(Source:
https://www.postoffice.co.uk/dam/jcr:f8877f55-cabc-4df0-a369-5a92c8c632b1/Card%20Account%20Terms%20&%20Conditions.pdf )
That suggests that the cards aren't 'made to order' (as normal bank cards are, with the holder's name an an expiry date on them) but are just generic numbered cards, taken from a pile in a Post Office branch, which can have anyone's account linked to them.
So, as far as I can tell, the answer to your primary question (about whether there's an expiry date) would appear to be 'No'.