The 24-year-old, who was born in South Carolina to two parents from Birmingham, switched his nationality from the United States last week.
In his first match for Britain, Crawford beat Ilya Ivashka in three sets to reach the second round of Australian Open qualifying.
“I would say I feel more British than American at times,” Crawford told BBC Sport after his 6-3 3-6 6-1 win on court seven at Melbourne Park.
“Obviously, I have grown up in the States but when you think of my heritage and what my family is all about, there’s not anyone from the States.
“I have a real passion for being with the Great Britain flag. Today hearing my name, and then playing for Great Britain, it gave me chills to be honest.”
He will get the chance to feel like that again when he faces Italian world number 195 Francesco Passaro on Thursday in the second of three qualifying rounds as he seeks to reach the main draw of the opening Grand Slam of the year, which starts on Sunday.
Crawford’s parents spent the first 30-odd years of their lives in Birmingham, before relocating to Germany where Oliver’s father Julian worked at a chemical company.
A job in a new office in South Carolina soon came up, and with his wife pregnant with Oliver, the couple moved to the city of Spartanburg.
Oliver, his younger sister and parents used to make regular trips back to Birmingham for Christmas or the summer holidays. Many uncles, aunts and cousins still live in the area, along with one surviving grandmother, and he has had plenty of opportunity to learn the family history.
“My grandfather is buried in Birmingham and whenever I go back I try and visit and see everything. My uncle is a big walker and so he [once] took us on a 15-mile walk all around Birmingham and saw their primary school, their middle school and everything growing up.”
Crawford, who is ranked 213 in the world and is now the British number nine, says playing for Britain has been in the back of his mind for years.
But why now?
“It just seemed to click at this stage – I think there’s a bunch of reasons,” he said.
“The camaraderie between all the boys is unbelievable. The culture they have built is wonderful. Obviously my family is in the UK, I love playing in Europe so having a base in London is going to be very good – and to have the opportunity to train with the calibre of players there, I think is second to none.”