The
Traveline South East journey planner includes TfL and national rail data, so that may be the best bet for enquiries like this.
Interestingly, for this journey it offered either train to Victoria then Underground or coach to Heathrow then Piccadilly Line which both take about the same time.
As far as I'm aware, any transport systems outside National Rail (e.g. London Underground / metro / tram systems) is not part of National Rail Enquiries' remit, although it does have approximate times built in for underground interchanges between the main line termini - although it struggles to deal with this outside the Underground's hours of operation (e.g. early Sunday mornings)