Tradescant

In the inscription on the tomb of the Tradescants in Lambeth churchyard, which it is proposed to restore as soon as possible, these two lines occur: "These famous antiquarians, that had been Both gardeners to the Rose and Lily queen." Can any of your readers inform me *when* the elder Tradescant came over to England, and when he was appointed royal gardener? Was it not in the reign of Elizabeth?