There is no standard progression; plus, you have to qualify what "having axels and doubles" means. There are doubles ... and then there are
doubles both in terms of quality and which ones are landed.
Frankly, I know a number of skaters who, after 10 years on the ice with privates all along, who still don't land a 2F or a 2LZ; but who have great spins, have passed all of their dances, moves in the field, etc. All skaters are different. I also know a few who took 8 or 9 years to get their doubles ... and then the triples came in a matter of months - whereas some of the kids who got their doubles in a few years never land a triple! Some skaters, weirdly enough, land doubles but still struggle with a single axel ...
Anyways, I would suggest that you not even think on these lines. Nothing drives a parent and a skater crazier than getting panicky about moving "too slowly". Some of the "fastest" moving skaters I know are now well behind skaters who moved more slowly; I've seen skaters come out of "nowhere" in terms of getting all of their jumps very quickly, and others who are top skaters who took significantly longer to get their jumps and spins.
There are too many variables in terms of basic talent, coaching, injuries, puberty, etc etc etc. And nobody can predict who will be a "star" skater based on progression at young ages. I look back at some of the kids who whipped through LTS ... and who are now far behind some of the kids who took longer to get through ...
Oh yes - my DD didn't land her axel until 10 years after she started LTS - coached privately by a top coach for the past 8 years, training 5 days a week, year around. Same week she landed her 2S for the first time. Now, she's close, 3 months later, to a 2L, has landed a 2F, and a 2T and nearly a 2LZ - which is interesting! She also has some of the best spins in our club ... a year after having the worst ones ... it just goes to show, she took longer than the other kids, but, is now catching up nicely - and in the interim, developed some fantastic skating skills and presentation skills ... my other kid had an axel that came and went for years ... same with the doubles ... now lands triples ...