Speech and language therapy.

I'd say that it's more important to focus on comprehension of language rather than expressive language (eg speech). Lots of ways to do this then think about helping the boys with ways to express themselves. Speech isn't the only way! I suggest that you contact the Royal Society of Speech and Language Therapy for info re local providers. You may need quite a specialist one.

There is something called the PAL program but it needs to be supervised by a speech therapist. One of my grandnephews has benefited from it but he attends a private special school. I realise that’s not very helpful for you but I’ll ask his mum if she knows of anything.

On thinking about it. I reckon what the problem here might be motivation. The ASD and ADHD don't predispose them to listening or even wanting to listen to others. It's understandable that his mum wants to home school but broadening their social horizons could demonstrate the need to communicate more effectively. Obviously, for the one with ASD the socialisation aspect is a problem. There was a very special school at one point for just such children but I can't for the life of me remember where it was. It was very good so it's bound to have been closed down!
I'd say PM me but I don't think that works at present.

As Aveline implies, more socialising could slowly help.
Have they always been home-schooled, or just since Covid?
When I was working as an Ed. Psych. we had a wonderful Speech and Language Unit which no longer exists.
I'm still in touch with the teacher in charge of the Unit and next time I speak to her will ask her advice,
She's my age (early 80s) and has health problems so is no longer active in this field but has a wealth of experience.