The Lymington and Keyhaven Marshes Nature Reserve is managed by Hampshire County Council. There is information on this reserve on their website here.
There are various routes as outlined on the image above. The Blue and Orange sections are suitable for cycling (gravel tracks with puddles when wet). Parking available at various locations including Keyhaven, Lower Pennington Lane, Lymington Bath Road Car Park, and some on road parking at Maiden Lane.
To the right you can see an example section of the seafront Blue/Red Circuit