Is pronouncing the word "schedule" as "shed-ule" only an upper class thing in the UK? Which pronunciation, "sked-ule" "or "shed-ule" is more faithful to the original etymology of the word, i.e. which came first?
[...] 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.
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.
The earliest English pronunciation of schedule is no longer used, as far as I know: it was something like /ˈsɛdjul/ (compare schism, which some still pronounce as “sizm”).
The OED says
In the 16th cent., both in French and English, the spellings scedule and schedule, imitating the contemporary forms of the Latin word, were used by a few writers. In French this fashion was transient, but in English schedule has been the regular spelling from the middle of the 17th cent. The original pronunciation /ˈsɛdjuːl/ continued in use long after the change in spelling; it is given in 1791 by Walker without alternative; in his second ed. (1797) he says that it is ‘too firmly fixed by custom to be altered’, though on theoretical grounds he would prefer either /ˈskɛdjuːl/, favoured by Kenrick, Perry, and Buchanan, or—‘if we follow the French’— /ˈʃɛdjuːl/. The latter he does not seem to have known either in actual use or as recommended by any orthoepist. Smart, however, in 1836 gives /ˈʃɛdjuːl/ in the body of his Dictionary without alternative, although in his introduction he says that as the word is of Greek origin the normal pronunciation would be with /sk/. Several later Dicts. recognize /ˈsɛdjuːl/ as permissible, but it is doubtful whether this was really justified by usage. In England the universal pronunciation at present seems to be with /ʃ/; in the U.S., the authority of Webster has secured general currency for /sk/.
It seems from this that /sk/ may be slightly earlier than /ʃ/, at least as a theoretical pronunciation for use in English, since Walker 1797 references earlier mentions of /sk/ by Kenrick, Perry, and Buchanan, while it was possibly his own idea to look at French (pronunciation rules?) for /ʃ/. (It's true that modern words in French spelled with sch- are pronounced /ʃ/, such as schéma, and as the OED mentions there are examples of "sch" being used in an obsolete spelling variant of this word in French—I found an example here from the early 1700s—but it’s not clear to me that Walker had any actual contact with a French speaker who used /ʃ/, or the "sch" spelling, in his time). Overall, it doesn't seem entirely clear to me which pronounciation, /sk/ or /ʃ/, can be considered to have become established earlier than the other.
The Latin source word seems to have a somewhat obscure etymology; however, I thought a brief discussion of it might provide some useful context for the various pronunciations.
It seems solidly established that the word comes from medieval Latin schedula/scedula, a diminutive form of a word scheda/scida meaning “a strip of papyrus”. This was apparently related in some fashion to a Greek word σχέδη/σχίδη, although the direction of transfer isn’t entirely clear:
Lat. schedula, a small leaf of paper ; dimin. of scheda, also scida (Cicero, Att. i. 20 fin.) a strip of papyrus-bark. β The Greek σχέδη, a tablet, leaf, may have been borrowed from Lat. scheda (or sceda ?) see Liddell; but we find also Greek σχίδη, a cleft piece of wood, a splint, which looks like the original of Latin scida. The difficulty is to know whether the Lat. word is original (from scid-, base of scindere) or borrowed (from Gr. σχίζειυ, to cleave). Either way, it is from √SKID, to cleave: cf. Skt. chhid, to cut.
– An Etymological Dictionary of the English Language, by the Rev. Walter W. Skeat, 1882. (I know this is an old source, but it’s the most complete description of the etymology that I could find out of the sources available to me.) If this etymology is correct, it would in fact mean that the the word is related to the English verb shed (as Henry said in a comment) but extremely distantly, so this wouldn't be expected to be relevant to the present-day English pronunciation.
In Greek, there is a difference between the pronunciation of κ (“c/k”) and χ (“ch”). The first corresponds to the voiceless velar plosive /k/; the second corresponded in early stages of Greek to an aspirated velar plosive /kʰ/, and in later and modern Greek corresponds to a voiceless velar fricative /x/.
In Latin, the digraph “ch” was originally used to represent χ in loans from Greek, and educated speakers likely pronounced it as /kʰ/, distinct from “c”. Later on, “ch” came to be used as an alternative to “c” in some native Latin words, such as “pulcher”; it is hypothesized that this might have been related to some phonetic development in these words. In even later Latin writing (e.g. in the medieval times), we see even more variation between “c” and “ch” (or “t” and “th”), which suggests that at some point, either one could be used to represent the same consonant /k/. Sometimes false etymologies from Greek also contributed to the use of “h” digraphs in the spelling of Latin words: the word “amarantus” came to be spelled “amaranthus” by association with the Greek word ἄνθος (anthus) “flower”.
So, regardless of the word’s origin, it seems likely that /sk/ was used in its Latin pronunciation at some point. Scholars have been aware of this for a while, and this is the basis for the /sk/ pronunciation in English.
The sound /k/ ended up being fronted before front vowels in the development from Latin to Romance languages. In French, the final outcome of this was /s/, and this pronunciation of the letter "c" was used even in learned words, which is why "cédule" was pronounced /sedyl/. This is the origin of the original pronunciation used in English with /s/ and no /k/.
In French, the digraph "ch" came to be used to represent the /ʃ/ sound resulting from another palatalization of /k/, and this caused many learned borrowings from Latin and Greek that were spelled with "ch", such as the word schéma that I mentioned earlier, to have pronunciations with /ʃ/ based on the spelling. (Since /*sʃ/ is not a possible word-initial consonant cluster in French, it's natural for sch- to be interpreted as /ʃ/). This seems to be what Walker (1797) viewed as the basis for a possible pronunciation with /ʃ/.
In addition to this, I have a suspicion that the graphical similarity of "sh" and "sch", and the use of "sch" for /ʃ/ in German, may have also reinforced the /ʃ/ pronunciation of this word in English.