I cannot provide a direct answer to your question, but I know a little bit about Old English.
The OE word is indeed "cirice". In OE the "c" was pronounced as a "tch" (as in MnE "church") in some cases and as a "k" in others. Generally this was the case before "i" and "e" ("cing" being an obvious exception).
If I recall correctly, this rule wasn't universal; so some dialects (possibly Northumbrian, too) may have pronounced it with a hard "k" in all places.
Furthermore, in German, which shares a common ancestor with Old English, the word is "Kirche". The "ch" is pronounced as the "ich" sound (not the one in Scottish "loch") and was sometimes used word-initially in old forms of German. In these word-initial positions, it was eventually replaced by "k" (leading to a frequent (mis)pronunciation of "China" as "KEE-nah" rather than "CHEE-nah").
So in other words, the phenomenon you're describing probably wasn't caused by a language change at a later time but was present in Old English dialects already. Not knowing more about OE dialects, I would assume that Northumbrian might have lacked the "ch" allomorph for "k" altogether.
