several decades earlier, say before World War I
Given the Brits had Congreve rockets in the 1810s, and were using them in the Crimean War, how big a stretch would it be for somebody to think improving the stability and range might be a good idea? Go from that to realizing black powder isn't enough, and just stumble on asphalt and perchlorate.
Given
that happens, how big a stretch is it for somebody to stumble on the idea of the motor having a cavity down the middle (which, AIUI, produces more thrust and a better burn)? And burning from the top, instead of the bottom (which, AIUI, produces a more stable flight)?
Getting to something akin to an SRB doesn't seem likely, because there's really no point to such a big lifter without nukes or man in space goals, but it does mean anything from
Hedgehog Mousetrap (or Weapon Alpha, or ASROC) to
Panzerfaust to MLRS (all the way up to LCT{R}) might be on the cards...
Edit:
IMO, most of the uses I imagine for solid rockets don't really need guidance systems, just decent stability (either fins or nozzles). They don't need to be steered, in general, to serve as artillery, ATW, or mine clearance systems (& arguably not as barrage AA, either). If they develop early enough to be practical AAMs, OTOH, they might--& that could be handled by radio command guidance, couldn't it? (Improving electronics enough for beam-riding, CW homing, or IR seems to want a POD of its own, unless we posit having better rockets has led to a desire, or need, for AAMs a
lot sooner than OTL.)