I don't know about "Old English", which is a language which absolutely nobody can read without years of study (In other words, it's much closer to, say, Swedish than actual English), but I do know that it was spelled that way until the eighteenth century in both America and GB, and in fact longer in GB. The spelling "complete" may also have been used before this point, yes, but the whole point is that English spelling was not completely standardized until very recently, and "compleat" was more common.
Dave? Hello Dave? Is Dave there? There's a block in your t'ilet.