- Truck spring wheel: http://encyclopedia.1914-1918-online.net/smw/images/thumb/a/...
- Bicycle spring wheel: http://www.oldbike.eu/museum/wp-content/uploads/2014/05/WW1-...
In regular environments (see: driving on paved road) they actively work against speed by reducing road contact due to tread gaps. That's why racing vehicles use smooth wheels:
http://www.theautochannel.com/news/2009/01/26/388620.1-lg.jp...
http://www.slowtwitch.com/articles/images/6/75576-largest_MI...
Plus most cars were still using slick (smooth) tires up until around the 1940s-1960s and (I don't believe) were fast enough to have a threat of hydroplaning.
ETA: Just remembered you were talking about the bike. Considering the background, it looks like it was used on dirt roads so if ever mud happened to be in the way, one could just carry the bike over. Dirt roads also don't usually require too much turning and are fairly sturdy, so there's little reason to believe skidding was a common problem.
Even if it were for sidewalk/road riding, you can easily get out of the way of puddles and slow down easier than cars.