I'm in the UK too. You can go via the GP, probably the best first move, if not helped enough or if there's a long wait for specialist assessment , you can look into getting assessment privately, costs more as you would need to pay. Have you mentioned this to him at all? Autism can affect our social interaction more strongly than our intellectual abilities. Surprised and disappointed if school doesn't recognise this. But I don't think their reaction is that unusual either.
Having said that, many on here are older and either ended up self diagnosing through personal research, and/or being diagnosed in later life. A middle way can be to find out plenty about high functioning autism yourself, and use this knowledge to assist and support your son, if you continue to see behaviours that are in line with your research.
Remember especially this is a neurological difference, not a deficit in itself. We are all uneven. He has some unevenness, just as we all do. Many of us, growing up unknowing and undiagnosed, had good careers, but may have benefited from understanding the root of our social difficulties sooner. But though it's good to offer appropriate support, the goal isn't to drive the child into masking his true feelings or wishes about social interaction.
He may be introverted, or need more than usual time to rest from a busy day with others who are probably mainly neurotypical. I like to decompress after a full day learning or at work, and still do not function well in unstructured social interaction, and tend to avoid such now, I go for semi structured interaction such as a class or volunteering. School may provide enough such interaction to satisfy him.