The Carlisle job didn't require radar and the Southend advert did state that training towards APS may be given to a suitable candidate, although I know that there were a number of applicants who were fully rated so perhaps it was filled by one of them.
You say you've been teaching for the last year, is that as an ATC Instructoror schools/colleges?
What have you done about finding a job? How many airports have you visited and how many ATC General Managers have you sold yourself to?
Have you enquired with NATS about direct entry?
The problem is that you appear to have had the requisite qualifications but you don't tell us what you've done to make the best of them.
In the last 18 months there have been at least 10 jobs that I know of (not all were advertised nationally) and I think you would have at least got an interview for, at least 7 of them would have carried the opportunity for training for the APS rating.
You are correct, not having the APS does make getting a job a bit harder but the world won't come to you.
Find out the name of the GMATS of every non NATS airport and write to them with your best CV, follow that up with a telephone call asking to visit the unit and to have an informal chat with the GMATS.
PM me if I can help.
Good luck.