The two met while studying painting at an art school in Cheltenham, and then returned to Barry's hometown of Strathaven, South Lanarkshire, to take over the family business – a path Claire never imagined they would follow.
"The signs were there," she said. "Barry once had a bakery van at college with Alexander Taylor printed on the side."
"It was a really new experience, but at the time I didn't expect the bakery to be part of the deal for us.
"In the end, circumstances worked out; we were in the right place at the right time."
When Barry's father became unable to run the business, the couple worked there for a while with his family, then eventually bought it from his parents.
But now, after more than 200 years in the Taylor family, Barry and Claire have decided it's time to move on and sell the bakery.
Alexander Taylor's has been in the heart of the South Lanarkshire town since its opening in 1820, yet both Claire and Barry said each generation has added their own unique style to meet the needs of the time.
In the early 19th century, Strathaven was a small, rural town.
Barry explained that his family were originally farmers before his two brothers moved away to start a business, and parish records detail the purchase of building materials.
It is said that production was initially low because when the bakery first opened, it was one of seven bakeries located in the town center.
It is the only bakery that remains today.
"In the Victorian era, my great-grandfather began to develop it more because there used to be a courtyard at the back," Barry said.
"Then in the 1920s, Strathaven became a town people would come to for day trips as the coaches which ran from Ayrshire to Edinburgh would stop there, so my family did a lot of catering towards that.
"There weren't catering companies so people relied on the bakeries."
Barry said he remembers his great-grandfather living in the same premises when he was a child in the 1970s, but his son, Alexander Taylor, died as a child, and the business was passed on to Barry's father.
"When I was of toddler age, I used to have a little wheelbarrow and I would bring stuff through to the shop from the bakehouse," Barry said.
"I don't really remember that but I have vivid memories of the bakehouse at different stages and the characters that worked there."
Even under Barry and Claire, the bakery has gone through several changes as the couple experimented with different recipes and new angles.
"We've came full circle, people are really appreciating the places that are unique to their area again and I think we've become an institution in Strathaven," Claire said.
"People are met here with a friendly face and we're part of a mix of things that make this town what it is."
At one point, Claire even organised pop-up art exhibitions in the building at the back of the bakery, which is used as a storeroom.
The bakery is now known for its sourdough, which Barry said he started baking before it became popular.
After decades of early mornings and long working hours, Barry and Claire soon hope to pass Alexander Taylor Bakery on to new owners and return to their creative roots as artists.
"We've always known that we can't do what we do forever and we feel like we have put a long shift in," Claire said.
"I've had people say 'I hope you never sell' and come up to me in absolute horror.
"But the main thing is we don't want to run out of enthusiasm."
A seventh generation of the Taylor family running the bakery could have been on the cards but the couple believe they are making the right decision and are doing what is best for the business.
Like Barry, who is said to have the bakery "in his blood", their two, now adult, children have grown up in and around the bakery.
Claire said: "In lockdown, our son and daughter worked with us and it was just the four of us - literally a family-run business.
"It's probably my favourite memory from the last 35 years working here.
"It was strangely really enjoyable and I feel proud of that time."
But Barry and Claire's son is at university and their daughter is a cyclist so Claire said she didn't think the bakery was their passion or part of their plans for the future.
"They have a great enthusiasm and respect for what we do but it's something I don't think they should feel burdened with," she said.
"In any other walk of life you wouldn't necessarily expect that you would do the same job as your parents have done.
"It is a unique thing for businesses that are family-run and what the Taylor family has done, it's quite an unusual thing for it to be in same family for so long."