Yup, Luther is Spot On
Technically, Luther's A und O (= English "A and O") is quite correct or as directly literal of a translation as one could get rendering the text into German (or English) without spelling out the two letters in view.
There are different Ancient Greek manuscripts and editions of the Apokalypsis Ioannou, "Apocalypse of John" (or the Book of Revelation), from which Luther could have been working, and, rather interestingly for the purposes of your Question, not a single one of them ever spells out the letter omega in this verse. Nor do they do so in the other couple of occurrences of the "alpha and omega" expression found in the Apokalypsis (such as 1:8 & 21:6) either. (At least of the manuscripts/editions that I have encountered; perhaps there are others that I don't know about. But even if any of them do or did [spell it out] it still changes none of my points here.)
For example, for the idiom in question in the verse you are looking at, a number of Westcott & Hort editions of the Greek, as well as the Nestle Greek New Testament (1904) and the Robinson/Pierpont Byzantine Majority Text each have
τὸ Ἄλφα καὶ τὸ Ὦ
(to Álpha kai to Ὦ), "the Alpha and the Ὦ".
The Stephanus Textus Receptus (1550), Scrivener's Textus Receptus (1894), the 1904 Greek Orthodox Church text, and the Tischendorf 8th Edition each have
τὸ Α καὶ τὸ Ω
(to Α kai to Ω), "the A and the Ω".
The name of the Greek letter omega actually means something like "Capital O," i.e.
O-Mega, literally the "Great/Big O." It is so named to distinguish it from the other O in the same alphabet, the letter omikron (Latinised as omicron), "small O," from
O-Mikron, the "little O." The Greek alphabet originally did not have the O-Mega, which was introduced in the 600s BC, and when it was added thereto, it was simply tacked onto the end of the alphabet, which is how it comes to be the last letter therein. Both letters are generally transliterated into German and English as O, o, essentially the same letter in all the three alphabets (German and English ultimately deriving theirs from the Greek one). Omega is sometimes transcribed as Ō, ō.
Up until recently all German Bibles, not just Luther's, have translated this expression as [das] A und [das] O, "[the] A and [the] O". The earliest one I've found which changes this up at all is the 1951 Schlachter translation, which seems to do so only hesitantly by spelling out the words in parentheses like so: das A(lpha) und das O(mega).
The Wiktionary article on A und O elaborates further on the topic and quotes one of the newer translations, saying that:
The biblical expression is not usually translated with these words
now. For example, in the Einheitsübersetzung verse 22:13 runs: “Ich
bin das Alpha und das Omega, der Erste und der Letzte, der Anfang und
Many recent translations have this rendition of the idiom, but the older A und O, from what I can see, appears to be a fairly common expression in German speech, just like "alpha and omega" is in English, meaning the same thing, and perhaps each deriving from translations of the Bible into their respective languages.
The same Wiktionary article defines A und O as a colloquialism for "that which is important; that which matters; a sine qua non" and renders an example:
Gute Kohle ist das A und O beim Grillen.
Good charcoal is the most important part of barbecuing. [Emphases in original.]
Redensarten.net also has an article on the topic, "Das A und O sein", offering several other such examples, of which here are a few, each with what GoogleTranslate automatically (and interestingly, I think) translated them to, further suggesting that, in common usage today, German A und O is simply equivalent to English "alpha and omega".
Gerade bei Allergien ist Sauberkeit im Haushalt das A und O.
Especially with allergies, cleanliness in the household is the alpha and omega.
Um die Sicherheit in deutschen Stadien zu erhöhen, ist Deeskalation das A und O.
To increase security in German stadiums, de-escalation is the be-all and end-all.
Doch so mancher Selbstdarsteller hat vergessen, dass die Wünsche der Kunden das A und O sein sollten.
But many a showman has forgotten that the wishes of the customers should be the alpha and omega.
The First and the... Last-ish
"A and Z" (or A und Z) is indeed sort of generally and roughly equivalent in meaning to
Α καὶ Ω, in both English and German, because English and German have the same basic alphabet. (Strictly speaking, at least since more recent times, German has a few additional special characters after Z.) While there is no German Bible I know of which translates the expression this way, at least two English ones do, namely, GOD'S WORD® Translation: "I am the A and the Z", and—perhaps the most colloquial mainstream English Bible translation of them all—Eugene Peterson's The Message: "I’m A to Z".
This might be too colloquial for some tastes, and to speak more strictly, to translate it this way is what would render it—based on the original meaning or the original alphabet in view—into saying something like the First and the Middle, because the Greek letter Z (zeta) is only the sixth letter in its own (24-letter) alphabet.
An English Afterword
By the way, in the 1380s-1390s, John Wycliffe published the first English translation of the entire Bible, roughly 130 years before Luther published his own German New Testament. This is how he rendered the portion of the verse you are quoting:
Y am alpha & oo, "I am alpha and oo".