Im convinced some riding schools are reluctant to have students cantering because it tires the horses out. Or if it's a group lesson it can be a bit hectic. Horses get excited and the rusk gies up.
At this stage you should have been cantering occasionally, if you're balanced enough to have an independent seat.
Can you book a private or semi- private lesson and say in advance that you'd like to work on your canter transitions?

Should say 'the risk goes up'

Yes you should be able to canter in each lesson by 16 months. There is nothing wrong with circles and serpentines, experienced riders will do these school movements each time they school their horses. You should also be cantering though, what does the instructor say when you ask about cantering? I’m wondering whether they have got the impression you are nervous?

How frequently have you been going? Once a week yes definitely, but a handful of times when you have funds, maybe not?

I'd certainly expect you to be cantering by now, if you have a balanced seat at trot with still hands and can maintain a decent rhythm then working on canter should be the next step, it shouldn't take 16 months.
If you can hack out it might be good to try a canter then, or ask specifically to work on your canter in your lesson. If they don't agree then ask them to explain why, they should have a good reason and help you to understand if tnere is anything that you can work on.

Definitely should be cantering by now if you're having weekly lessons for 16 months! For sure speak to the instructor

Yes absolutely you should be cantering by this stage. I think you need a very serious conversation with your teacher (have you had the same teacher for the whole 16 months and do you have private or group lessons?) and unless they can get you cantering within the next couple of lessons I'd move schools TBH, canter is tricky for a novice rider but not 16 months tricky (absent any significant physical issue or big problems with nerves) so I'd be really interested in what their 'excuse' is - if it's a control of the horse or balance problem they could at least pop you onto the lunge and get you to hold onto the saddle or a neckstrap and canter a couple of circles so you can as a minimum feel what it's like - but obviously it's easier for schools to keep you pottering around in walk and trot and never progressing any further.

I'd also want to find out once you are cantering what would their plan then be, you should be doing no stirrups work, pole work, prep for jumping, going out hacking etc IMO with the length of time you've been riding, that is not unreasonable you'll find with most schools (unless again you are very nervous or struggling with your fitness or whatever). My worry would be if they're happy for you to have paid for 16 months worth of lessons and not have you really progress, they'll get you cantering then think 'well she should be happy now' and you'll stagnate again at the level of doing 20 mins worth of circles and a quick canter either way at the end forevermore because that's what suits the school/teacher rather than what you are capable of. Like others have said school exercises are of course very important but there's no point practicing the same things over and over with no real understanding of what you're trying to achieve or what your goals should be. A good teacher should be able to easily tell you what their aims for you within (a) the individual session (b) the next couple of weeks and (c) longer term are and what you need to do to get there. That can include things you should be doing off the horse too like working on your fitness, core strength, flexibility etc or improving your confidence or whatever as well as actual riding skills development. Not all teachers will set that out for you unasked but they def should be happy to discuss...

You should be jumping by 16 months of weekly lessons let alone cantering .

You should definitely be cantering by now. It took me a year or very very sporadic lessons before I cantered - I started in June 2021.

I had fortnightly-ish lessons for about four months, then a gap of four months as my instructor was on maternity, then about three months of mostly weekly lessons, then I started canter work.

I've since galloped and cantered down the beach and started pole work too ☺️

This!

I would consider looking for a new yard. I was having lessons somewhere local. Initially I thought they were great when I was totally new but then realised I was getting a different teacher each time and normally a different horse which meant we had to spend the first part of the lesson just getting to know each other and I never really developed.

I switched yards to a much smaller place with only one instructor and I ride generally the same two horses. We have a plan for each lesson and do all sorts of interesting things- poles, jumping, different transition type exercises. I've been going there for less than a year. My next plan is working toward x country! So to answer your question yes something has gone wrong somewhere if you're not cantering after 16 months.

While I agree with others re. Cantering by now, please don’t think of doing serpentines etc as ‘going back’.