As others have said, actually having an exam certificate is pretty much never important. The grades are usually there as a guide so teachers or parents know which ensemble is appropriate to send which child to. They may not even ask, you say (having seen it's grade 1+ and your DS is good enough) my son would like to join the school orchestra, they say great bring your violin on Monday.
Let your DC settle in with the new teacher with no pressure from your part to do exams. Teacher planning an exam straight away would already be a bad sign in my view. They need to get to know each other, teacher needs to explore strengths and weaknesses and find out how your DC learns. If they try to start exam prep straight off they won't know how fast your DC will pick things up and are fairly likely to end up with the wrong timing again - so either DC will be ready too soon and get bored, or will again not be ready in time for the planned sitting and have to defer. If your DS got most of the way to preparing an exam with the old teacher, it may be best to count that one as done, get settled with new teacher and pick up at a later grade in due course. You don't need to do all grades.
How you know your DC is progressing. Exams aside, you should be hearing them learning new music on a regular basis, and hearing each piece get better from week to week. There should be progression in the music they're learning - at early stages they often have a teaching book that they progress through, as the music gets more advanced it may be more targetted particular pieces that address different skills. A good teacher will also be setting them appropriate scales and/or exercises even if there isn't currently an exam in view.
With a young child preparing for their first exam, I think you should be very in touch with what they have to do and see that they're practising for each component. If you can be in lessons, do. Or look at your DC's lesson notes to make sure they're remembering to practice all the things they should be. Learn enough yourself to understand what's going on. I'm musical so that was easier for me - but when DC were doing gymnastics badges or swimming badges I looked up enough to understand what they had to tick off so i could encourage appropriately, and got DC to explain the bits I didn't understand.