I would choose a Mahagony colored stain, Olympic has a nice one.

You went for a very English look so I would paint it white to match your trim etc. Which you said you don’t want…

I understand painting would look best but that is the side of the house that is always damp and it would be green with mildew soon enough.

I usually like deck flooring to look like the floor inside so what is taht ? It will not matter wht you do for the rest it waill always be filled with mildew on thta side of the house so I agree no paint just a good stain and dark too I cannot not imagine it white but then I do not like white with brick . So what color are the floor sinside that is IMO the way to go.

Reddish/brown stain to echo whatever bricks on the garden wall have brown in them--a bit darker than this (about the color of the darker boards in this photo).

Patricia,
Hardwood dark walnut color but the floor that runs into the deck is deep red brick off that is black and white mosaic tile. No red. I hear the country music song about a double wide and red wood deck. That is not the look I am going for. But the darker walnut or dark gray to match the mortar is what I am pointing towards. Thanks for the input.

They make solid wood stains in pretty much any color, don't get transparent or semi-transparent stain. It's still a wood stain, not paint, but gives the same effect....no natural wood color showing through.

I would do the walnut the dark gray IMO does not suit the house

Here's a reddish brown deck with a red brick house. Since you also have a red brick wall, the color connects all three areas. Add color with plants and colorful outdoor cushions and umbrella.