Crocus's Mark Fane is museum chair and Capital Gardens' Colin Campbell-Preston is vice-chair.
Director Christopher Woodward has raised £100,000 for the museum through sponsored swims in the English Channel, Bosphorus and Straits of Gibraltar, and is planning a 30km Arctic swim.
Christopher Woodward in Tradescant gallery
Many artworks have been added to the collection of garden objects and archives.
Archives include those of garden designers and plantspeople Penelope Hobhouse, John Brookes, Beth Chatto and Andrew Lawson. Paper records of modern gardeners such as Russell Page, Hugh Johnson and Joy Larkcom are being added.
Inside there are seven new galleries, with a Tradescant's Orchard exhibition from the collection of Elias Ashmole, sponsored by Tiptree, ongoing. Planned exhibitions include Humphrey Repton and Cedric Morris paintings. Burle Marx could be a subject of the future.
There is a new bronze extension with a Pearson's Tradescant garden with plants from Crug Farm among others, plant hunter John Tradescant's tomb and learning spaces for 90 schools a year. Woodward said the garden was an opportunity to see one that is properly planted, maintained and planted.
Elements of the new museum include: The Ark Gallery of 25 items from Tradescant's collection from the Ashmolean Museum; Ashmole's tomb; opening of The Tower; and learning studios. There is also a cafe and workspace.
Woodward said the near thousand-year-old church will now house many items the museum, founded in 1977 by Rosemary Nicholson, could not display before. Star items include a Repton red book, Gertrude Jekyll collection, William Robinson's cloak and Harold Gillman's Black Gardener painting. Some 1,200 garden objects such as ancient watering cans and Alan Titchmarsh's Gardener's World jumper are on display.
Woodward said two-thirds of children in central London live in houses without gardens but they will be able to study plant biology at the museum. A food learning officer will also be employed.
Improvements meant now was the first time in 1,000 years it was warm enough to sit comfortably in the building, which is also used for weddings and other events, Woodward added. He hopes this will help make the museum self-sufficient. Woodward is also planning more acquistions.
He said: "Why would someone come here rather than visiting a garden? We're putting on something a garden can't give."