You know, the easiest thing to suggest is simply to buy a bigger tire. 1.25" tires on a bike that came with 2" knobbies is a substantial reduction in tire size. When I bought my brother tires for his upright bike, I wouldn't go below 1.5" because I felt that the ride would become to rough.

Schwalbe rate their own tires with a scale on their site. The Big Apple and the Big Ben only get a 4/5 for speed.
Big Apple HS 430 | Schwalbe North America
Big Ben | Schwalbe North America

Also, at a quick glance, I don't believe either tires comes in smaller size than 2".

Their "luxury" tire is the Marathon Supreme, it's smallest size is 1.6" (also comes in a 2") -
Marathon Supreme HS 382 | Schwalbe North America

But it gets a 5/5 for speed. Basically it's more expensive, but does everything better than the previous two - faster, lower weight, better grip, maybe a better ride as well.

If you're looking for a cheaper tire with an emphasis on speed, the Marathon Racer looks good to (note that it's not actually a race tire in any way, but it rated tops for speed) -
Marathon Racer HS 429 | Schwalbe North America

It comes in 1.5 and 1.75.

(All sizes are for 26" wheels which the OP said they had).