NHS Lothian said all parents of child patients would be contacted directly.
Three-year-old Teddy was born with Shwachman diamond syndrome, which means he has a higher risk of cancer. He undergoes blood tests every three months and bone marrow biopsies every year to monitor his condition.
He has been receiving his treatment at the hospital’s current home in the Sciennes area of Edinburgh until now.
The Sick Children’s Hospital had been due to move to its new £150m building in Little France, on the outskirts of the city, on Tuesday.
The announcement was made on Thursday. The following day, Danielle, from Balerno, received a letter which said that Teddy was to have blood tests in the new building at the beginning of August.
The 28-year-old told the BBC Scotland news website: “When we received this letter we phoned the haematology department at the old hospital but we just got a secretary’s voicemail and our messages have never been returned, so we don’t know what to do.
“I feel we have been left in the dark and I feel very uncertain about what is happening.
“He is also due his bone marrow biopsy, so not knowing has left me feeling very upset and annoyed.
“I feel frustrated. They shouldn’t have released the new opening dates until they had done all the checks as its left families waiting for different tests and scans and nobody knows what’s going on.”
Analysis by Lisa Summers, BBC Scotland health correspondent
There is no quick fix to the ventilation problem discovered at the new Sick Children’s Hospital.
Regulations state that in critical care wards the air change rate must be ten times an hour.
It appears the discovery made last week was that the current system is not powerful enough to do that. Engineers and experts will have to get working on a solution but it is likely to take months to complete.
Then you have the problem of when to move a hospital? Summer has been favoured because it is quieter.
People are on holiday so less planned procedures and in general, there is less sickness.
The weather too, can help. Do you really want to start the enormous logistical challenge of a hospital move at the coldest, darkest time of year?
Danielle said her son has been admitted to hospital several times recently.
She said that on each occasion they had noticed “a dramatic decrease” in things that made it a nicer atmosphere for Teddy, such as toys and fish.
“He doesn’t understand what’s going on so it is very important that these things are there for the children,” she added.
“At first I thought the hospital would only take a few more weeks to open, but now I’m hearing it could be next year so the toys and fish need to be returned to the old hospital.”
NHS Lothian stressed that it could not comment on individual cases but promised parents would be given information about any new arrangements.
Chief officer for acute services Jacquie Campbell said; “We recognise this is a worrying time for parents and carers who have appointments at the new Royal Hospital for Children and Young People and would like to apologise to them.
“We can reassure them that they will be contacted directly by the team at the existing Royal Hospital for Sick Children to confirm arrangements for their child’s appointment.”
Ms Campbell said the priority had been children with appointments this month.
“People whose appointments are in July are being contacted by phone in the first instance,” she added.
“Those who have appointments in August onwards will be contacted by letter. Every effort is being made to retain the same appointment date and time wherever possible.”