Posts Not Paginating

I am copying a clients site over, so I have exported and reinported
the posts etc, and the theme into their new site.

Unfortunately the problem being that this is not my theme, im copying
it over for a client and its displaying correctly on their old site on
a lower version of wordpress

For some reason your jquery is broken as @TomJNowell suggested. The question here is, you said that everything is exactly the same on the old site and the new site and you have copied everything excatly to the new site.

Here are a couple of things to try here

One, flush your permalink by visiting the permalinks page and just clicking update

Secondly, it might be that you missed url paths in the db. Download and install Velvet Blues Update URL’s, add

http://www.cyclezoneuk.com/

in the old URL field and

http://www.griffscyclelab.co.uk/

in the new url field, select all options and update. BUT PLEASE, make a back-up of your db first before attempting this. There is no roll back. A good idea here is to flush your permalinks again when you are done

Thirdly, have a look in the theme if they did not hard code any URL’s in to some of the templates. Apart from that, everything should work on the new site. I really don’t think that worpress version will have any effect here, you are only upgrading from 3.6 to 3.9, so I would see no issue here

because it says twentytwelve in the content, rather than the theme
name, its not pulling through.

That would make no difference. This is only the text domain name which is actually used for localization of strings. The only effect this will have is that your translations won’t work, ie, your translation for a string won’t be returned. You should always have this the same as the text domain specified in style.css though. But like I said, this has got nothing to do with your actual problem here

EDIT

I’ve tested the widget plugin pagination plugin, and there is a bug in the plugin. Deactivate the plugin. It is also compatible up to v3.4.2. So deactivate and delete the plugin. I would also deactivate all plugins, and reactivate them one by one to check if this problem is plugin related