No, I think not, unless the route via Weybridge is within three miles of the shortest route from Staines to London, which I don't think it is.
Other than direct trains, the shortest route and any routes within 3 miles of the shortest route, tickets from Staines to London are only valid on map WX on this file.
If you really want the ability to go both ways then a season from Egham is valid via Weybridge as well and costs £3.10/week more. Starting short at Staines is allowed on a season ticket. If going via Weybridge you would be making two journeys: Staines to Egham and Egham to London.
Edit: Not quick enough.