Question: I've recently started to do dedicated sight-reading exercises. As I make progress I can either move on towards more difficult pieces or stay at the same level while increasing the speed. Currently I'm doing the former while playing rather slow (around 60bpm).

Will that approach also improve my speed in the long run or should I practice for speed separately?

I'm playing the piano, but I think the question applies to other instruments, too.

Answer: 
Its worth noting that as you develop as a player you will become better at sight-reading simply because you become a better player. 

But by definition you cannot attempt to sight-read a piece more than once can you? After that its not really sight-reading is it. 

So @alephzero has made a really significant point that is worth repeating: If you want to use sight-reading as a skill then you need to be able to read the pieces at the intended tempo. Anything else is just practicing the instrument.

You need to find lots of music and just try playing it. Start with things that are simple. Move on when you find yourself managing that level of piece easily at first sight. Be critical and be honest about what you managed and what you didn't. That will give you an insight into where you need to strengthen your technique.

Good luck.
Sight reading is a place where you should always try to push yourself to harder music. Speed can be practiced in different ways, but the benefit of sight reading really comes from your brain being forced to focus on seeing and processing the notation. Part of the notation is playing at the indicated speed. Sight reading a Largo can be quite different as you sub-divide to stay in time. 

So practice playing faster in other ways, separately, and push the "harder music" in the sight reading.