I'm pretty sure Shakespeare used the subjunctive. I'm not sure about Chaucer and I haven't a clue whether it is used in Beowulf. Has the subjunctive always existed in English, even as far back as Old English? If not, roughly when did it start being used?
The subjunctive mood has always existed in English. It in the Old English of Beowulf and in the Middle English of Chaucer and the English of Shakespeare.
The question suggests a misconception that the subjunctive mood is something that was acquired at some stage in the development of English. In fact the subjunctive mood is a feature of Indo-European languages, including the Germanic ancestor of Old English, even if its usage changed during the development of the language into Modern English, where it is much less used than previously.
Web references to the subjunctive in different ancestors of English
Here swice and scolde illustrate the subjunctive mood (N.J.Engberg).
It is ful lasse harm to lete hym pace, Than he shende alle the servantz in the place.
Here shende is subjunctive, used after than.
Be thou a spirit of health or goblin damn'd, Bring with thee airs from heaven or blasts from hell, Be thy intents wicked or charitable, Thou comest us such a questionable shape that I will speak to thee ...
In this extract from Hamlet the italicized words are in the subjunctive mood.
The subjunctive is used to "express situations that are not yet realized or hypothetical and is typically used for what is imagined, hoped for, demanded or expected. [It has mostly disappeared] because most of the functions are covered by the modals could, would, and should." OED
In English it is almost indistinguishable from the indicative "except in the third person singular where the normal indicative -s ending is absent ... and in the verb to be...." OED
He face rather than faces as in ...it is recommended that he face a tribunal. And in the verb to be : If I were rich instead of if I was rich.