Forest were confirmed by the EFL against a rival claim from Stoke City, who claim to have been founded in 1863 – two years before Forest switched to playing football instead of shinney (a form of hockey).
Stoke's claim (promulgated by the inclusion of the legend '1863' on the club's badge) is supported by FourFourTwo magazine, which purported to list "The 40 oldest clubs in the Football League" following Notts County's relegation.
But there is no documentary evidence to support this claim. Essentially, it does appear that a club was formed in 1863 by a group of Old Carthusians who were working as apprentices at the North Staffordshire Railway works in Stoke; but there is no evidence to link this club to Stoke Ramblers, whose formation in 1868 is the earliest event to which Stoke City can trace its history.
In any case, Stoke went of business some time in the Edwardian era, as confirmed by Wikipedia: "Continuous history from reformation in 1908 following the wind up of the original club."
The Nottinghamshire Live website reported on 9 May 2019 (four days after Notts County's relegation) that there were doubts about Stoke's claim and that the EFL were investigating. Forest's own website reported on the following day (10 May) that the EFL had settled the dispute in their favour, and Wikipedia (on the page previously cited) agrees: "The EFL confirmed that Nottingham Forest are now the current oldest member club due to the lack of evidence to support Stoke City's claim of having formed in 1863."
© Haydn Thompson 2019