This is from another topic in the British Isles section of the forum r.e. postmarks but is also relevant here:
I am not an expert, can try to answer these questions based on my knowledge!
Do I put the postcards in an envelope addressed to the relevant centre, with each one also stamped and addressed to it’s ongoing destination?
Yes, exactly this! Not sure if there is a limit on the number you can send in one envelope though… @Maddymail I don’t know if you can help?
Do I post them on the date I want the postmark to be, or do I need to factor in travel times?
The postmarks are usually valid for a window of time, except permanent postmarks, but you can check this in the Royal Mail hand stamp bulletin here. Looks like you might have already read this though 
Is the code for the relevant postmark part of the address or must it be written on the postcard itself?
Write the code clearly on a piece of paper and put it in the envelope with the cards. I’d also write the code clearly on the back of the envelope, just in case!
Update: Just copying and pasting this straight from the bulletin: To ensure prompt handling please mark the outer cover ‘Special Handstamp’ and state clearly the serial number(s) of the handstamp(s) required.