When Matthew died in 1814, he was still living in London working with the Publishers of his great works A Voyage to Terra Australis. So his remains were buried in London. The location of Flinders' grave in St. James's Garden, a former burial ground, was unknown by the mid-19th century. 

From 1789 to 1853, the site was used as an overflow cemetery for St. James’ Church, in Piccadilly. Later alterations to the churchyard obliterated his grave. 
Back in 1852, Flinders’ sister-in-law, Isabella Tyler, visited the St. James’ cemetery and reported that his grave was gone. In a letter written years later, Flinders’ daughter reported that her aunt “found the churchyard remodeled, and quantities of tombstones and graves with their contents had been carted away as rubbish, among them that of my unfortunate father, thus pursued by disaster after death as in life.” 

Research shows Flinders’ remains were almost certainly moved to an unmarked grave (or possibly just dumped) just a little to the east of their original spot, as part of the modern Euston station.

It was finally rediscovered 200 years later in January 2019 when his remains were unearthed during an archaeological excavation preceding a major new railway development. 

Archaeologists working with High-Speed 2, a planned railway linking London to Birmingham, Manchester and Leeds, planned to exhume up to 61,000 bodies from the former burial ground located next to Euston Station in north London. Flinders, was believed to be buried somewhere near the station. 
A lead plate bearing his name clearly identified his resting place, much to the excitement of Donington and Australia!  The race to bring him home was on...