Hi!
When will be this type of functionality be available on-wiki? I have a
Well, since it has REST endpoint (https://www.mediawiki.org/wiki/Wikidata_query_service/User_Manual#SPARQL_end...) that can be called from JS or PHP or anything in between, technically anytime beginning now :) but I guess it depends on the meaning of "available". It is still beta, so some changes and tweaks and stability challenges possible, but otherwise it is ready for use.
That said, I'm not sure how exactly wikis such as the English Wikipedia may be using this engine - that's something we need to start figuring out, now that we have the engine.
vague memory that some type of query service similar to this one was/is intended to be used to allow arbitrary access to Wikidata data from/on Wikimedia wikis such as the English Wikipedia.
Well, technically query engine is not direct access to Wikidata data - it's a copy of the data, not the actual data, and also for some accesses - such as single property/label value retrieval - it may be more efficient to use the direct access APIs that Wikidata is enabling right now. But for more complex requests I think that is the engine that would be supporting that.