111 are not medically trained, they often make errors, they work from algorithms which don't cover all eventualities. 111 do not know what the A&E doctors and nurses will do.
Of course it could be nothing, a virus/ behavioural thing etc.
But if he has symptoms of type 1 diabetes then he really needs a simple blood test (fingerprick or normal blood test) to check his blood sugar, and a urine test. The blood test is imperative, the urine test alone wouldn't be sufficient. It could be done in a GP surgery or hospital. This will tell you if he is becoming severely ill from type 1 diabetes, as his sugar would be high and his body would be making something called ketones which would show in his urine. In this case the child could be drinking and weeing loads, feeling unwell, drowsy, tummy pain, sweet smelling breath, losing weight. Not necessarily all of the above. This is an emergency, people can and do die from this. Not to be overly dramatic, but if your DC seems unwell I would take this seriously and go to A&E/ similar.
If you do decide to go to A&E/ walk in centre or similar, at triage say that your DC has symptoms of type 1 diabetes and list them, don't start by talking about the sleeping.
If you decide to stick with your GP's plan, tbh I would call them back and clarify the type 2 diabetes thing. A normal weight 5 year old will not have type 2 diabetes. If your GP is thinking this I would seek another GP as that is bonkers. Hopefully it is a misunderstanding though.