Leeds City Museum is great! They have a little 'Toddler Town' on the 2nd floor amongst everything else to look at on the other floors! Depending on which day you go, they also have craft sessions, but there will be a timetable up showing the dates/times....plus it's free! :) The City Library also has a toddler area with the Tiny Totley Studio with a slide, toyes etc...the kids love it!
They do story and rhyme at the library on Thursday mornings. And I think a song time on a different day too. There’s also a place called Hidden village like role play, maybe a bit old for him (and a bit pricey) but could be worth a look. I quite like having a little potter about the market too, lots to see and often things on in the events space.
Tiny Totley at Leeds library. A bit further out, but still walkable/short bus ride is Sparks at Thackray Medical Museum. My 17mo loves it x