The car on the left in the old photo definitely doesn't match the modern photo of the 10hp Wolseley.
The clue is the radiator - the red Wolseley has the radiator tank back at the dashboard (ie just in front of the driver's feet) with tubes running horizontally around the sides and front of the bonnet. You can see four of the tubes joining the tank. This is an older style of radiator.
The car on the left in the old photo has a more modern style of radiator, entirely at the front of the car.
Furthermore, compare the rakes of the steering columns. It's almost vertical in both cars in the old photo, compared to a rake of around 45 - 50 degrees in the red car.
These are details that don't change with the coachbuilder.
The car on the left in the old photo is a very distinguished vehicle. All car owners of that time had to be wealthy, but this car would have been sold only to the wealthiest of buyers.
I'm not making a positive identification, but it is similar to the Panhard et Levassor seen here (and said to be of 1904):
https://www.gracesguide.co.uk/File:Im10LBVCR-541-Pan.jpgLook at the shape of the radiator and the rake of the steering column.