The terms you are using are ambiguous and confusing, as is the situation.

You said that you have "26 x 2.15" wheels on your old Raleigh, but that they are not right for the frame. What isn't right? Do the brakes touch the rim correctly? Are the tires too wide? Are the wheels too tall (too big of diameter)? Are you looking for skinnier tires that would fit better?

Wheels have a diameter and rim width.
The diameter can be labeled as 26", 700C, 27", 26 x 1 3/8", 26 x 1 1/2",
or with ISO sizes like 559 (aka mountain bike 26") or 622 (aka road bike 700c).
Very old Raleighs used a couple uncommon rim diameters, so you need to confirm the diameter on your wheels.
The rim width isn't as important; you can fit wide tires on narrow rims.

Tires also have a diameter and width. The diameter needs to match the rim. 559 tires fit on 559 rims. 1.5" is 38 mm, or 2.125 is 54 mm. Both sizes are made in the 559 diameter, common to mountain bikes. The 38 mm tire won't be as tall but should generally roll a little faster than the 54 mm tire, which will have greater cushion and more traction.

Fitting smaller (35-40 mm) tires with a smooth tread to an old mountain bike is common for city riding. But make sure you get the correct diameter tires to match the rim diameter.