Did you report the missing bag to the CO handling desk in the T4 luggage hall?

I hope you did, and if so, I am sure they'll get the bag to you soon. In my experience, their handling of lost/mising bags in the UK is excellent.

If not, and from your first post, I guess you did not (a big mistake I am afraid to say), then you are in the hands of CO (on the assumption they are the originating carrier).

Either way, I do not see much point in trying to return to LHR, as I doubt you'll get much help there, and as you have found out, it will be an expensive trip for you from Teeside.

Although frustrating, my best advice is to await reply to your emails and/or keep rying to get through on the phone to the LHR CO handling desk.