The Royal Botanic Gardens at Kew, London and Edinburgh each have some records relating to their former staff and students, which can be searched by their own archivists.
At Kew (@Kewgardens) their staff records start in 1841 (pre-1841have not survived) and apprenticeship/student records start in 1902.
The records are arranged in date order therefore it is important that you have an approx. date of employment before you make enquires about the research. Your enquiry should be made in writing including as the employee’s full name, the dates when he or she might have been working at Kew, and their occupation, if known. Records may be incomplete, so it may not be possible to confirm your ancestor worked at Kew. https://www.kew.org/explore-our-collections/library-art-archives
The Edinburgh (@thebotanics) gardens have online links to alphabetic lists of probationers, Leith Walk gardeners and those who belongs to the guild. https://www.rbge.org.uk/science-and-conservation/library-and-archives/archives/family-history. There is also a page commemorating the workers who were killed in WW1. https://stories.rbge.org.uk/archives/category/learning/ww1. In house researchers can do further research for which they may charge, so it is essential to contact them first with as much information as possible.