The stock wheels on the Sports are 650A which is 26 by 1 3/8 which is 590mm while a 26 inch mtb rim is smaller at 559 and are not interchangeable.
The improvement if you go from the chromed steel rims to some CR18's will be astounding and being that you live where you do, wet braking ability is a must have.
I ride a Phillip's 20 that has been super modified and after switching to alloy wheels my braking was still poor because of the weak stock brakes (the steel brakes on your Raleigh are much better) and despite using Kool Stops and then I brazed on canti / v brake mounts and with cantis the braking got much better but I still wanted more so I installed V brakes and now I can do mad skids.
This stopping power would not be possible with steel wheels.
A cool idea would be to lave a new CR18 to the rear hub and perhaps lace a new Sturmey Archer drum hub to the front or get a generator hub and lace that to a CR18 so that you can power a headlight.
The 40 rear and 32 spoke front is typical of Raleighs of this era and they moved to a 36/36 lacing in the seventies
If you drop a 20 or 22 tooth cog in the back your gearing issues will be addressed in that you will have a low that will get you some steeper hills at the cost of a top gear which was always set too high on these bicycles anyways.
Who wants to hurry when you are riding such a beautiful old bike ?
I build wheels for a living but live too far away to help... you should be able to find a reputable and experienced wheel builder who can lace new rims to your stock hubs and the front hub on the Raleigh is spaced at 95mm which means modern 100mm hubs will not fit unless you adjust the fork blades and machine the dropouts slightly for what will be a wider axle.
Invest in some decent stainless spokes as these will make you happier in the long run and will really make the bike sparkle for years to come.
If you go the route of getting the CR18's and new spokes and have these wheels built around your existing hubs you are looking at $180.00 - $200.00 for the parts and build fee which is not bad when you consider the value of having such nice hand built wheels.