Disheartened and feel like giving up after son's appointment

My 2 year old son had his first face to face appointment with the paediatrician just before Christmas. He's already had a few appointments since 2019, including a telephone consultation with the paediatrician and a face to face with the nurse. 

Originally we were told by the paediatrician that she would be seeing him every 3 months, with the nurse seeing him alternate months. She said she may even diagnose him as early as March, she just wanted to see if the nursery picked anything up first (he's starting this month). 

However, at the appointment he mainly stuck with me and my husband. He didn't want to move around and play, he was just really clingy. He eventually done a small puzzle, but because he kept looking at me she said that was a good sign. She was concerned about his 'weird finger movements' as she called it and him spinning. But nothing else concerned her and she said as there was nothing obvious she thinks she'll wait around another year before seeing him, he's 3 in September so she said maybe the end of this year. 

She kept saying it's fine, he's really young, it could be a phase etc. I told her he's been hand flapping since 8-9 months old so I doubt it's a phase. She just didn't answer. I know he's young but isn't the whole point that the earlier the diagnosis the better? 

I honestly feel like screaming or crying. My husband and I have both been so upset, angry and disheartened since the appointment. I feel like we'll be lucky to get him diagnosed before school now. My daughter and I are also being assessed and I feel like if they've missed signs in him, who has really obvious traits, then what hope do we have. 

We are really struggling with him but because he didn't act like a performing monkey during a 50 minute appointment and do things on demand then we can't get anywhere. The only help we can get before a diagnosis is links to websites and leaflets! He has a speech delay, rocks violently, hand flaps, hits his head off walls, slaps himself, wakes up for between 4-5 hours every night after not going to sleep until 10-11pm. He's started hitting us and throwing things. I feel so upset and don't know what to do. 

Is there anything we can do? Can we ask for a second opinion by a different paediatrician? Surely this isn't a fair assessment to judge how a child acts during one appointment for less than an hour? The nurse told me on the phone it does come down to how they act during the appointment unfortunately and the next appointment could go the same way. This can't be fair, I'm considering saving to go private but don't know if the NHS would accept it.