The SnakeGrid coordinates (E/N) inherit the datum of the geographic coordinates from which they are derived. Since these geographic coordinates are ultimately derived from GNSS surveys, the GRS80 ellipsoidal parameters are assumed.
In practice, this means that SnakeGrid can be used with any modern datum that is related to WGS84 or ITRF. In Europe this will probably be ETRF89, in the USA it may be NAD83.
Formally, the SnakeGrid coordinates of a project such as Crossrail, for example, are described as “SnakeGrid XRail09 coordinates on the ETRF89 datum”. However, XRail09 is usually a sufficient identifier.