 Fellowes was born in Edinburgh the third son of William Fellowes, physician-extraordinary to the prince regent - he was elder brother to the naval captain Thomas Fellowes.He was educated at Rugby School before spending time in the British Army's medical department, becoming surgeon's mate in June 1794.He then took his medical degree at Peterhouse College and Gonville and Caius College, along with London lectures from George Fordyce and Andrew Marshall and time in Edinburgh, finally graduating MD in 1803.