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Please read the book " the late talker" you'll find many answers.
Sending love from the US! My son was in a very similar situation at that age. We had him evaluated and he wasn’t interested in participating so he scored abysmally and the evaluators were basically convinced we were doomed. He’s now 5 and about to graduate from speech therapy. He talks all day and it’s amazing. I know not every kid will have that trajectory but it did happen for us and hopefully it will happen for your kiddo too. We didn’t get a diagnosis, though autism and adhd were mentioned as possibilities. We did private speech therapy, which was the available option for us in the US. We tried a couple different therapists before we found one who clicked with us. If your son doesn’t speak yet have you tried teaching him baby sign language? We did that and it served as a bridge to spoken language. Reached the character limit… hang on…
On the advice of his speech therapist we also got him an ipad with a speech program on it to use as an assistive communication device. There were words and phrases loaded on it so he could use it to say, for example, “I want cookie” or whatever. He was talking a bit when we got him this, and it was especially helpful when he was trying to talk but we couldn’t understand. Sometimes he would sigh in exasperation and say “blue ipad?” And we’d hand him the device and he’d find the words he was trying to say. The device and the sign language got him used to the idea of having a voice and communicating with us (there’s research to support this claim) and helped us get him to explore speaking. He also benefitted immensely from a speech therapist who knew about gestalt language processing, which is something you can look into if it ever feels relevant to your situation.
My son started talking a little around 2 1/2, made slow and steady progress through age 3, sped up from 4 onward, and now at 5 he’s full speed ahead. He still has some age appropriate challenges around some sounds but is making good progress with them. The other say someone remarked upon how well he speaks and I nearly fainted. I’m so sorry your family has to navigate this challenge. It’s so hard and frustrating and terrifying. Don’t give up and remember to take time to relax and enjoy your kiddo and all the wonderful things that he does do. I wish you great success and that your kiddo will soon talk your ears off. Please feel free to reach out to me if you think that could be helpful.
Sorry two more things — Him pointing, waving, gesturing, and experiencing frustration at not being able to communicate are all huge rays of hope. They show that he understands the concept of expressive communication and he wants to engage in it but for whatever reason he can’t right now. These factors make me think he might enjoy and embrace alternative communication methods like signing and/or an assistive device like I mentioned. Also him being smart and knowing numbers, letters, animals, etc. is a huge win! It shows that his brain has plenty of horsepower and is able to retain and compile lots of information. He honestly sounds so much like my kid did at that age that I genuinely have so much hope for you guys. Hang in there mom! You’re doing so great in a hard situation and your kid will benefit immensely from your dedication and love.
@Rosie we are in the UK
I'm sure they do have early intervention program in UK as well. You can talk to the pediatrician to get evaluated for speech delay
I don't know where you live has Early Intervention services or not. Have you tried anything like that