Hi and welcome,
I'd suggest Richmond Park > Roehampton Gate > Priory Lane > Upper Richmond Road > Queens Ride > Lower Richmond Road > Putney Bridge > King's Road > Victoria > The Mall > Marlborough Road > St. James St > Piccadilly.
That's the easiest route but I'd use the Embankment from Edith Grove all the way to Trafalgar Sq via Westminster Sq as it'll be faster with less stops.