How can I upsize downloaded photos?

the short answer no

start with the biggest pic you can can

some site will actually show you a larger pic if you click on it or right click view image and then save the bigger one

irfanview is a good picture viewer and there are some resize/resample options that are easy like a couple of mouse clicks.

I usually double it and see how it looks and if its still ok sharpen it then double it again

if not i go 50-75% sharpen and go again

and as far as wallpaper you can always tile them

Unfortunately, if it's a raster image that has many colors and no hard lines, not even vectoring it will help to resize it.

If it was simple shapes with hardlines, you could trace it with an vector program (e.g. Adobe Illustrator; CorelDraw; PSP vector tool, etc.), or recreate it in a paint program.

Otherwise, increasing the size will cause pixelization.