Ds is an extreme case. Speech delays are fairly common and most will go on to gain functional speech. Ds was non verbal till nearly 9. It was actually an alternative communication method which led to the speech he now has. He's now 15, has some speech (it's complicated), he mainly uses a high tech communication aid. He communicates effectively.
Unfortunately my experience of nhs salt is it's a mess. We've had a lot of success with private salt.
Alternative communication methods are very much misunderstood. The goal is communication and they very much do help support speech development if used correctly (they often aren't). For the park for ds I would use a picture of the actual park. It depends what your little one is okay with. At 2 we needed printed out visuals, now we use a picture on our phone or he chooses which park himself via his communication aid.
Whether you use pecs, object to reference, makaton, high tech aac the idea is to get them used to some sort of communication and support speech development. Hand leading is helpful but doesn't promote independence. It's great for letting you know what he wants but he is 100% reliant on you. That's fine at this age but trust me it's huge issue as they grow older, it's incredibly limited. With alternative methods you're essentially supporting the verbal word with a symbol/picture/sign. They can't say it currently but you absolutely can. Modelling is exceptionally important. For example if they wanted a biscuit, you could use a picture they could bring to you, point to etc, you'd have to model how they do this, it's something that takes time to learn. Many years in our situation. Once they've indicated this is what they want or you have modelled it you say 'biscuit' and give them time to make a vocal attempt if they are able to. Never withhold anything, the goal is to teach them a more positive way of communicating without distressing them at all. Ds showed zero interest in makaton, very little in pecs really. It's finding what works for them.
Requests are obviously far easier. For other situations such as you mention at gymnastics. Did he understand that's were he was going? For ds he would behave like this because he was anxious, he needs to know what's going on. It sounds like he settled into it and struggled with the transition of the activity ending. This was a huge issue for us. The thing that eventually worked was signing 'one more finished', it's the only sign he's ever taken any notice of. It's essentially a warning sign to tell him the activity is finishing. I usually do this 5-10 mins before the end so he can get used to the idea. I then say finished and show him visually what's happening next. He didn't understand verbally so it's another way of including him, a way he understands.
Waiting is a difficult skill at 2. We implemented a visual waiting symbol, it wasn't successful till much later but we had significant receptive language issues until age 6. At this stage the goal is some sort of functional communication to hopefully reduce his need to display challenging behaviours. The more communication ds has gained in whatever form the more the challenging behaviours have reduced. Ultimately he just wants want he wants and if he's got a positive way to get that there's no need to behave negatively.
I hope this makes some sense! I've waffled.