I posed this question to @OdeonCinemas, @OdeonHelp and @ApolloVictoria on Twitter and received the following replies from each:
Provided it doesn't create a trip hazard, or get in the way then there shouldn't be a problem.
-@OdeonCinemas (source)
As long as it doesn't block any aisles you will be fine :)
-@OdeonHelp (source)
So small bags in the Odeon appear to be fine. And the Apollo has a cloakroom:
Hi James we have a cloakroom you can put your bag in!!
-@ApolloVictoria (source)
Update: Having now visited, I can confirm that my small backpack wasn't even given a second glance at either venue.
9 year update: I've been fortunate to have had the opportunity to visit Wicked at the Apollo Victoria multiple times over the last year - bags are still fine! I don't recall whether it was a thing on my first visit, but there is now a bag search. Unsure what they're looking for (professional video gear maybe?), but I've always been fine with the typical "day out in London" stuff.