It is possible to ride from Tunks Park to the Spit and follow council signs all the way.  But its all on roads and its very hilly.  I did it early on a Sunday morning and saw no cars, this will not be the case any other time.  Thats the downside, the upside is it is very elevated and the views when you get them are fantastic. It also connects with the Spit which means you are able to join to the Manly tracks.  It was much better heading from the Spit to Cammeray than the other direction as you get to ride on the one-way Wonga rd and the blocked off Bay St.  Best way would be ferry to Manly and ride back to St Leonards station and catch the train there.

So take care and try it on a Sunday.  Unless you are a really strong rider, its not a great road for commuting.
15km return trip. Really hilly. Many experienced riders switch to the busy Spit Road and run down the hill. Not my cup a tea.

See pics here >>

See the Very Light Grey trail in Google Maps here on the Sydney Map >>

See also this article on the eastern side of Mosman …

The Zoo to the Spit Recreational Trail and the Bail-Out Commuter Trail