Conway was born in Taunton, England. He qualifies to play for Scotland through his paternal grandfather, who was born in Stirling.
In March 2024, Conway said of his plans for UEFA Euro 2024 if unselected for the Scotland playing squad, “Scott Murray funnily enough is the kitman at Bristol City. He’s driving from Bristol to Germany with his brother. If I’m not going I said I’d be jumping in with him. It’ll be class." He was reported, "Through his late grandfather Danny, who hailed from Stirling, and dad Joe he is a passionate Scotland fan."