OK 1st thing several "26"," 590 is one size rim different from MTB 559.

Take the EFFORT to MEASURE the RIM RADIUS the above numbers are Diameters . r= 1/2 D

Take the center of the brake track on the rim .. to the hub axle.

British EA3 is the 590, common in past on 3 speed English bikes if so the tire size you had before is the one You get again , they are a Practical Size .
rims steel?

My Little Brompton Has the same tire width, 35mm, but a 10" smaller diameter Rim..

If you want to use wide MTB tires you need to Buy a different Bike