The 13-month-old has a dangerously enlarged heart and his family are clinging to hope he will be able to have a heart transplant to save his life.
But Oscar will only be able to receive a transplant from a donor a couple of years older than him.
His parents are calling on peers to support Max's Law – which will change the rules on organ donation from an opt in to an opt out system – when it's debated in the House of Lord's on Friday.
Oscar condition deteriorated rapidly earlier this year when the powerful drug keeping his heart pumping nearly failed.
His dose of milrinone, which relaxes the blood vessels in the heart, was increased and he has since stabilised, but his family know it's only a temporary fix.
Oscar's mum Abbie Burkmar, from Bournemouth, told The Sun Online: "When he’s doing well he does seem like any other happy child.
"On his good days he is really good and we are hopeful the he will get through it, but then we have days where he’s not so good and we start to panic about him deteriorating and things like that.
"But overall, so far, things have been relatively smooth and we are hoping they stay like that."
Little Oscar was diagnosed with dilated cardiomyopathy in January – a condition which causes the heart to become enlarged and unable to pump blood – after a midwife detected a heart murmur the day after he was born.
In June his health took a turn for the worse and he was transferred to Great Ormond Street hospital in London for specialist care.
He stabilised and was allowed home over the summer, but in September his family were given the devastating hews he was suffering heart failure and he was referred back to Great Ormond Street.
In a Facebook post earlier this month Abbie said: "To start with his journey was pretty smooth and uneventful.
"His heart actually slightly improved after being on medication for a month and he seemed like any other happy, chunky baby. Then at the end of May he deteriorated.
"His heart worsened and Oscar wasn’t coping. He was taken to intensive care at Southampton General Hospital and the doctors told us there was a good chance he wouldn’t improve and therefore they were referring us to Great Ormond Street hospital so that he could have a heart transplant assessment.
"Thankfully after a few weeks of intensive care and amazing medication Oscar turned a corner.
"His heart showed no improvement but his body was starting to cope again and we were able to go home and stay off of the transplant list."
But the improvement was short-lived and Oscar's condition started to deteriorate again, with his family taking him in and out of hospital.
"He underwent a blood transfusion and was put back on stronger medication meaning he has to stay in hospital," Abbie continued.
"On October 16, nine months after being diagnosed, Oscar was placed on the waiting list for a new heart.
"The doctors don’t think his heart will improve and they’re not sure if we’ll be able to go home before his transplant. If they do get us home, there’s a good chance we’ll be back again as Oscars heart is very weak."
Oscar is being kept alive by a drug called milrinone, which relaxes blood vessels and allows blood to flow more easily through his veins, reducing the pressure on his struggling heart.
But in late October the drug stopped working and Oscar's heart started to give up.
Doctor's worked tirelessly save his life, increasing his dose of the drug to help his heart.
His condition has stabilised and he is doing much better since his dose was increased, but for now he is being kept in Great Ormond Street hospital where doctors can keep a close eye on him.
As often as possible he is visited by his dad Josh Dunkley and four-year-old brother Jack.
"Oscar has been doing very well the past few days, responding well to the increased dose of milrinone," Abbie wrote on Facebook on November 4.
"For now the plan is to change nothing. Keep his meds the same keep his NG tube in for his feeds and take all the pressure off of his heart and hope that we don’t see anymore deterioration.
"All in all a good outcome for now.
"So proud of our boy for staying strong and bouncing back from the awful way he was in last week."
Oscar has the same condition as 10-year-old Max Johnson, whose case stated the debate around Max's Law.
A GIFT OF LIFE
Organ donation is giving an organ or tissue to help someone who needs a transplant.
Transplants can save or greatly enhance the lives of other people.
But this relies on donors and their families agreeing to donate their organ or tissue.
There are two types of organ donations; when a person is living and when a person has died.
In death, most organs can be donated.
While a donor is still living, they can choose to donate a kidney, a small section of their liver, discarded bone from a hip or knee replacement and also their amniotic membrane.
Doctors can only use organs and tissues from a registered donor with the families consent after they die.
So if you are a donor, make sure you have discussed your wishes with family and friends.
To register to donate your organs visit organdonation.nhs.uk
If the law is passed it will mean the UK operates on an opt out organ donation system rather than an opt in – it will be presumed people want to donate their organs unless they opt out.
Max finally had a heart transplant last year and now Oscar is desperately waiting for the same chance at life.
Currently, if you wish to donate your organs, you need to join the NHS Organ Donor Register or tell a relative or close friend that you wish to donate.
To register, fill out a quick form on the NHS website, which takes around two minutes to complete.
Max's Law will mean that everyone over 18 will be presumed they want to donate their organs for transplant when they die, unless they actively register their wish not to.
Ministers say the changes will be in place within two years.
Under-18s, people with limited mental capacity and others who have not lived in England for at least a year prior to their death will be exempt from the scheme.
Those who do not wish to donate can record this on the NHS register either online, by phone or on an app to be released by the end of the year.
The legislation, which was introduced in Parliament in July 2017.
Abbie is urging everyone to consider organ donation – something that she admits she hadn't thought of until Oscar's diagnosis.
"If you’re a parent think ahead about if the worst is to ever happen to your child or yourself what would you want to happen so that if, god forbid, if it ever was to happen you have already made that decision," she told The Sun Online.
"If you’re struggling to think about whether you do want to sign up flip the situation and think about if it was your child that needed an organ, what would you do?
"If you would accept one you should really be signing up too."
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