I've never heard of a "1 speed tyre", but it is possible this is a harder durometer rubber to resist wear if the rider does a skid-stop, at the cost of being less grippy. It should still fit your rim the same.
The only likely gotcha here is whether both 26" tyres are the same size.
Check the ETRTO code on the sidewall. It will be in the format 559-56 or 56-559. The bigger number has to be identical, but the other number is the width and your new tyre will be around 46. Both are in millimetres.
You probably don't need new tubes for this relatively small change in tyre width.