An inquest into the death was opened and adjourned on Friday, but the results of a post-mortem have so far proven inconclusive. The couple claim the wasted time could have saved their daughter's life.
Mr Linforth said: "After about 20 minutes waiting at the surgery she got worse, her lips were blue, she was really hot and was having trouble breathing.
"I went up about four or five times to ask to be seen. They took her into the nurse's room, but by that point her heart had stopped beating."
Miss Collins said: "This should never happen again to any other family. I will never get over this. I want to get Lucie's story out there and try to stop it happening again.
"She was a healthy, vibrant, lively character full of fight, and her life was just chucked away that morning. If people had just opened their eyes they would have seen just how poorly she was. Time was precious and it was wasted sat waiting."
Lucie died on October 5 this year after she had been unwell at her home in Bedford the night before. A police investigation has been launched, although the death is not being treated as suspicious.
More than a million people have visited a Facebook campaign page dedicated to Lucie, and which has 30,000 'likes'.
Miss Collins said she was now campaigning to ensure sick, vulnerable children and elderly patients are given priority in GPs waiting rooms.
She said: "I want to see patients who are seriously poorly in prioritised in the waiting room. Nobody should be left to die because they are, fourth, sixth or 16th on the list. I will be fighting for Lucie's legacy for the rest of my life - something has to come out of the tragedy."
Doctor Fiona Sim, medical director for NHS Bedfordshire and Luton, said: "This is a tragic death of a little girl and our thoughts and condolences are with the family.
"The family must be going through terrible turmoil at this devastating time. It is vital that the evidence and facts are gathered so we can all understand what actually happened in this tragic case.
"As this is an unexpected death, there is an ongoing coroner's enquiry and we are providing every assistance as well as working with the practice to carry out an internal investigation.
"Local NHS services are supporting the family and we wish to offer our full support in any way that we can to help all involved during such a difficult time."

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