And charities say this leaves "great concern" for their future health - some patients show early markers of eye, heart and kidney disease.
Data from England and Wales suggests more than 1,000 new cases were recorded in people under-25 last year.
But overall blood sugar control in this group has improved, the study shows.
Type 1 diabetes often appears in childhood. It is caused by the immune system destroying cells in the pancreas, leaving patients unable to control their blood sugar.
This is different to the far more common type 2 diabetes, which is largely due to an unhealthy lifestyle.
The Royal College of Paediatrics and Child Health (RCPCH) looked at figures from young people's diabetes units across England and Wales in 2013-14.
Its analysis shows that 26,867 children and young people with diabetes were recorded last year, compared with 25,221 in 2012/13.
But just 16% underwent all seven annual health checks that are recommended to monitor their blood sugar control and any complications.
Persistently high blood sugar levels can lead to serious problems, including heart disease and kidney disorders.
The figures suggest:
Barbara Young, Diabetes UK chief executive, said there was evidence that children were experiencing problems with their eyesight or kidneys before they reached their 18th birthday.
She said: "This is tragic and we have great concerns for their future health if they are already showing signs of complications at such a young age.
"There is an urgent need for the NHS to make the pace of improvement quicker."
Dr Justin Warner, of the RCPCH, said: "This is the first time we have been able to collect and analyse the data on early warning signs and levels are higher than we would expect.
"There is some good news though - we can see from the audit that blood sugar control is getting better across England and Wales.
"We hope this will have a knock-on effect, reducing these early warning signs too."
The study also looked at type 2 diabetes, which accounts for 5% of all cases of diabetes in children and young people.
Many young people with this form of diabetes live in deprived areas of England and Wales.
A "worryingly high" number of children with type 1 diabetes have warning signs of long-term health complications, including blindness, a report says.