I'm pleased to say I can give a successfull update to this inquiry:
I've just had a phone call from Molly at Stockton Bereavement Services and she has confirmed that Swires is indeed buried in the old part of Thornaby Cemetery, section A, plot no.E16. There's nothing to say whether or not there is a headstone but, as diggerman2 intimated, she says I should phone to make an appointment for a member of staff to show me the grave. However, as it would mean travelling up and staying overnight, I've no idea when this will take place. The main thing is that, thanks to you all and to the excellent help from Stockton B.S., I now know where my gtgrandfather is layed to rest.
Many thanks,
David