Cerebral palsy will not stop you doing what you want with your life. That was the message Diana Webster gave her daughter Victoria from her earliest days. Then, at the age of 10, Victoria announced that she wanted to become a doctor – and Diana admits that even her iron nerve faltered just a little.

"I thought, oh my goodness. She was the most determined child you could imagine, and once she set her heart on something, that was it. But becoming a doctor … I thought, this time she's bitten off more than she can chew."

Having a speech impediment and an uneven gait wasn't going to deter Victoria, as Diana quickly realised. Victoria passed her exams, applied to medical school and refused to be daunted by the growing pile of rejection letters. Eventually, she was offered a place at a top university in Sweden.

Not only did Victoria, 47, qualify as a doctor, she went on to become a hospital consultant. What's more, she works in the accident and emergency department.

As we chat in a cafe, Diana, 82, explains that people tend to assume that Victoria's disabilities mean she is intellectually, as well as physically, challenged. "When we're out, people often speak only to me, leaving Victoria out of the conversation. Or you get people who see her reading and say how brilliant that she can read. We know what they're thinking: blurred speech equals a blurred mind."

Sometimes, Diana fantasises about one of these dismissive, judgmental people keeling over with a heart attack. How splendid it would be to say, "Let my daughter through – she's a doctor" and have them make way for the person they thought was the village idiot, but who turns out to be a highly skilled hospital consultant. Diana looks a bit embarrassed that she's had such a thought: but who could blame her?

The Websters' story begins in hospital, where it also ends, because Diana's account of giving birth is almost as shocking as Victoria's account of her struggle to become a doctor. In 1965, Diana and her husband, Mike, were teaching at Helsinki University. It had been an easy pregnancy but the labour was long – much too long, she now knows. As it was Good Friday, her obstetrician was on holiday. "These days they would do a caesarean and everything would probably have been fine," says Diana.

But they didn't. When the baby was born she didn't cry – and when the doctor asked for the baby's name, for an emergency baptism, Diana feared the worst.

But Victoria survived and, at first, it seemed she might have emerged from her difficult birth unscathed. As she started walking, however, she seemed to fall over more than other babies – and when she embraced her parents, she didn't purse her lips, but gave wet, puppy-dog kisses. A visit to a specialist resulted in a diagnosis of what was then called spasticity: during the delivery, Victoria's brain had been starved of oxygen. She would, said the doctor, have unclear, slow speech; she would probably always be clumsy and have difficulty walking; and there would be plenty of other things she'd turn out not to be able to do either.

But the family – now joined by a boy, John, two years his sister's junior – refused to be daunted. "We simply thought, Victoria can do anything any other child can do – we're not going to limit her life or her choices," says Diana.

At school, though, Victoria knew she was different. "We were still living in Finland, so I thought maybe it was because we were English," she says. "I was being teased by the other children and thought that must be the reason. My perception of myself was then – and is now – that I'm completely normal. Intellectually, I know that I have a handicap: but it's not something I'm aware of all the time."

In 1979, Mike died of a brain haemorrhage. Victoria, then 14, was in the years to come much influenced by a family friend, who was a physician. "He always said being a doctor would be better for me than being a nurse because I'd need more fine motor skills for nursing," says Victoria. "I knew I'd need experience before medical school, so I took short-term jobs in a hospital and in an old people's home. That strengthened my resolve – I knew medicine was what I could do and be good at."

Victoria applied to medical schools in Britain and the Nordic countries – the family briefly returned to Britain after Mike's death, but resettled in Finland – and was eventually offered a place at the Karolinska Institute in Stockholm, Sweden's leading centre for medical education and research. But it didn't take long to realise that this wasn't the pot of gold at the end of the rainbow.

Victoria was one of 140 students in her intake, the only one with an evident disability. "I'd got in thanks to my qualifications, without an interview," she says. "In the first few days, I noticed curious glances, so I decided to speak to the course leader, to explain, for example, that I had poor handwriting and that it often got worse under stress. I didn't think that should matter very much – after all, doctors are renowned for their illegible handwriting!"

The course leader shocked Victoria. "She said she was very grateful to me because my arrival at the institute had shown there was a problem with the intake system. She said I'd shown her that they had to change things so that people like me didn't get in to study medicine in the future.

"She said people would be frightened of me and wouldn't be able to trust me. She advised me to give up my place, and said they'd admit another student from the waiting list in my place."

At that moment, says Victoria, all her years of fighting prejudice seemed to crystallise. "I told her I wouldn't go. I said I was quite prepared to accept that there might come a time when I'd have to admit that I couldn't continue as a doctor, but I'd have to see it for myself. I was determined to go on doing my best to achieve what was being asked of me, and not to give up until that became impossible."

Over the next few years, Victoria studied hard, acquired experience in different areas of medicine and eventually graduated. Then came opportunities to work at hospitals in the US and Europe including, for two years in the early 1990s, a spell at the Derriford in Plymouth. While she was there, working in the accident and emergency department, that she decided this was the branch of medicine she wanted to specialise in. "There are so many aspects to A&E – you have to work across so many specialisms and you really see how medical care really can make a difference," she says.

By this stage, she had perfected a way of introducing herself to patients and their families. "I realised that what was important was to say something to every single person I encountered, to explain that I have a speech impediment, and it's something I was born with. It's important because that tells the patient that I'm aware of it, and I'm aware that they might wonder. It also flags up to them that I was employed with this handicap – it's not something that's happened to me since I was hired – so the hospital clearly believes I can do my job despite it."

In 18 years of medical practice, says Victoria, only two or three patients have said they didn't want her to treat them. "And those times have been in very stressful situations where I was up against it – if I'm confident, things are always OK," she says.

At present, she is an A&E consultant at the Haartman hospital in Helsinki, where she sees between 60 and 90 patients over a 24-hour period. She is happy there but would not rule out the possibility of one day working back in Britain – her family still has a house in Cornwall, though Diana and John and his family are, like Victoria, based in Finland.

Diana hopes her daughter's story will inspire every parent who has a child with a disability to believe that anything is possible. "Too often, the emphasis is on can't from the outset. But nobody can say with absolute certainty what a person with a disability can or will be able to do in the future – that's what I hope Victoria's story proves."

• So Many Everests by Diana and Victoria Webster, is published by Lion Hudson, £8.99. To order a copy for £7.19, including free UK p&p, go to theguardian.com/bookshop or call 0330 333 6846