Hampton Court Park
This 750 acre park is open to the public and sits next to Hampton Court Palace and Bushy Park in south London. It's great for an afternoon stroll with long tree lined avenues running along the Long Water canal in the centre of the park. At the end of the canal, the Golden Jubilee Fountain shoots spectacular 30 metre jets of water into the air. Deer, descended from Henry VIII's original herd roam around the rest of the park freely. Adjacent to the park you will find Bushy Park which is a great option if you'd like to continue your outing.
You could also cross Kingston Bridge and try the Richmond to Kingston River Walk or the Kingston to Hampton Court River Walk which use the Thames Path National Trail. You can visit the pretty Teddington Lock and Canbury Gardens on the route.