I was in a similar position to you 9 years ago with DS at the same age as yours. I was referred by the HV with an unspecified wait for SALT and I ended up paying for private weekly sessions with him whilst waiting for the referral to kick in.
I am so glad I did because after 8 weeks, my private SALT completed an assessment outlining her concerns and I am sure this made a difference getting DS seen quicker.
Once we were in, he had fantastic help weekly from SALT, OT and specialist playgroup (sadly closed now due to cuts), learned Makaton and went to a mainstream nursery at a children's centre where they signed alongside speech. I did parents' courses in Makaton too.
I remember at the time wishing DS had been deaf, because at least then, he could get help/hearing aids/learn BSL. In our case, people couldn't tell me when he would develop speech and I worried very much that he just wouldn't.
He was diagnosed with Speech and Language Disorder but managed in mainstream primary and is now looking forward to starting high school with no extra help. He now has just a lisp, but I am sure that without all the help he would have many more problems, including behavioral linked to his frustration. He has no SEN, it was just the speech and language.
I know there are arguments about buying your way up waiting lists etc etc, but at the time, I wanted to do the best for my child. NHS SALTS are fabulous, there's just not enough of them and 4 months is a lot to a 2 year old, IMO.
Also, completely ignore anyone who tells you that Einstein didn't speak until he was 11 or whatever. Einstein was an extraordinary freak of nature.