Some of the questions I might ask would depend on what/how I wanted to play. If, for instance, I wanted to play jazz, I would look for a teacher who would teach improvisation and how to play by ear. If I wanted to eventually play in a wind band or concert band, I would look for a teacher who taught by a more classical route - notation, sight reading skills etc.
When you say your present teacher is "qualified" in sax, what type of qualification is it? Some qualifications place more emphasis on academic study, some on performance skills, and vice versa. It's something to bear in mind, although it's often been mooted on here several times that qualifications are less important than teaching skills.
I would try not to commit myself to a teacher until I'd tested the waters. All my teachers have offered a free meeting, during which they suss out whether they want to take you on and you suss out whether they're the right teacher for you. Not all teachers do this - some offer a "consultation lesson" which you might be expected to pay for.
If I had to find a new teacher, I would probably say something along the lines that I wasn't sure if I could commit to regular lessons and could I just have, say, three lessons to see how I got on. That way there's no awkwardness if you discover they're not the right teacher for you, and you'll have a better idea of what you're looking for next time. Not all teachers may agree to this though, and you run the risk of losing your slot if someone who is willing to commit long term wants it in the meantime.
On a practical level, you might want to find out whether they follow academic terms or whether they teach all year round. The loss of several weeks lessons during the summer may mean that your progress is checked until the start of the new term. If you want to do exams, the closing date for the winter session is very shortly after the start of the autumn term and if you haven't had lessons all summer, it's more difficult to assess whether you'd be ready for the winter session exam, and it may mean you delaying until the spring session.
Did you use www.musicteachers.co.uk to find a teacher? On the whole it's probably reliable, although it's easy for someone to claim qualifications that they don't have, and this has happened on occasion. If in doubt, look for evidence of framed certificates, or ask some questions about where they studied, what their qualification involved etc. It's very hard to do this, which is perhaps why nobody does. I would particularly warm to a teacher who volunteered information about their qualifications, teaching plan, attitude to exams etc.
You could also have a look at www.sax.co.uk
which has teacher lists. Or get in touch with some concert or wind bands via www.amateurorchestras.org.uk
and see if any of their saxophonists are teachers.
Good luck with this, trimmy. It's a minefield finding a good teacher who suits you, but worth it in the end!