The British pronunciation of the word "schedule"

Solution 1:

From Etymonline:

[...] the modern British pronunciation ("shed-yul") is from French influence, while the U.S. pronunciation ("sked-yul") is from the practice of Webster, and is based on the Greek original.

To answer your question directly, the modern spelling appeared in 15c as a throwback to the Latin schedula. Google's pronunciation of Latin schedula sounds like skeh-doo-la to me. This is closer to the typical US pronunciation.

The pronunciation guides I checked list the UK pronunciation as simply "UK". Whether it has a class distinction isn't something I can answer.

Solution 2:

I've never been aware of a class distinction aspect of shed-yul vs. sked-yul.

I will note that the American pronunciation has made its way back over the pond to some extent. If that is due to the influence of film and TV (which I imagine it is) then it could well vary over different demographics.