As far as catching up, it's just so hard to say what kind of time scale will be involved.
Ds1 had speech problems when he was a toddler - for the longest time he was only speaking key words - no sentences, just specific key words to get his point across (and meltdowns when he couldn't). It made nursery a challenge sometimes as they spent a fair bit of time trying to figure out what he was trying to tell them.
And then suddenly he went into overdrive and repeated everything he heard verbatim - but got it all mixed up. So if he wanted a biscuit, he would say "Would you like a biscuit?" because he couldn't mentally switch from what WE asked him to asking for himself by saying "I would like a biscuit" or any variation of that. All his conversations became very confusing because he would latch on to a word or phrase he'd heard before and interject that whole sentence into the conversation, even if it didn't make sense. It was like talking to someone that was switching the dials on the television repeatedly IYSWIM.
Now he's 8yo and talks constantly. Very very fast and struggles with anything that is not literal. If you tell him that it's raining cats and dogs, he will go look out the window and tell you it isn't. But he can have conversations now - they're just very quick and very much on his level of cooperation, which can be quite interesting.
He's still a work in progress. Not sure if that helps though.
Our hearing test at that age wasn't fine (failed on the tympanography but passed the main test). Hearing can fluctuate so rechecking is a good idea.
As far as catching up; we were at a similar level of speech at that age and still not caught up at five. But this is not really the place to ask that question. Quite a lot of children do take off and are fine by three or four. The parents of those children won't be on this board to answer that question.
Moderate delay can be scoring 34 where the advantage is 38-58. Doesn't tell you much unless you know what age range the score puts them in or their percentile rank. A percentile rank of about 8% is a moderate delay - but maybe the top of the rank is 30% say, I would ask what the age delay is.
My dc was language delayed at two, he came on a lot between four and five, discharged from salt at six, caught up in my eyes around age seven or so but he is still quirky and hates being asked questions. Tbh it is impossible to predict outcomes at this age. Also ask whether her receptive language is delayed
It really depends babies what the underlying issue is. DD was moderatly delayed at that age but despite therapy the gap got bigger. We now have a severe spreech and language delay about the 0.2nd percentile. Catching up will probably never happennon our casem