Analysis of the Greek Text
According to Joseph Henry Thayer, on the word πᾶς,1
III. with negatives;
- οὐ πᾶς, not every one.
- πᾶς οὐ (where οὐ belongs to the verb), no one, none, see οὐ, 2 p. 460; πᾶς μή (so that μή must be joined to the verb), no one, none, in final sentences, Jn. 3:15 sq.; 6:39; 12:46; 1 Co. 1:29; w. an impv. Eph. 4:29 (1 Macc. 5:42); πᾶς … οὐ μή w. the aor. subjunc. (see μή, IV. 2), Rev. 18:22.
In addition, on the word οὐ, he wrote,2
plur. οὐ πάντες, not all, Mt. 19:11; Ro. 9:6; 10:16; οὐ πᾶσα σάρξ, not every kind of flesh, 1 Co. 15:39; οὐ παντὶ τῷ λαῷ, not to all the people, Acts 10:41; on the other hand, when οὐ is joined to the verb, πᾶς … οὐ must be rendered no one, no, (as in Hebrew, now לא … כָּל, now כָּל … לא; cf. Winer, Lex. Hebr. et Chald. p. 513 sq.): Lk. 1:37; Eph. 5:5; 1 Jn. 2:21; Rev. 22:3;
To summarize:
On the one hand, if the Greek text had stated «οὐ πάντες μὲν κοιμηθησόμεθα πάντες δὲ ἀλλαγησόμεθα», then it would be translated as "Indeed, not everyone shall sleep, yet everyone shall be changed." Here, οὐ πάντες (οὐ immediately preceding πάντες, rather than the verb κοιμηθησόμεθα) is understood as "not all," as in "some shall sleep." It would not mean "none, no one," as in "no one shall sleep (i.e., die)."
On the other hand, since the Greek text states «πάντες μὲν οὐ κοιμηθησόμεθα πάντες δὲ ἀλλαγησόμεθα», with οὐ immediately preceding the verb κοιμηθησόμεθα rather than πάντες, it would be translated as "Indeed, no one shall sleep, but everyone shall be changed." It could also be translated as "all shall not sleep" as long as it is understood as "no one shall sleep" rather than "some shall not sleep."
In his commentary on 1 Cor. 15:51, Heinrich August Wilhelm Meyer wrote,3
This interpretation alone, according to which οὐ, in conformity with the quite ordinary use of it (comp. immediately οὐ δύναται, 1 Corinthians 15:50), changes the conception of the word before which it stands into its opposite (Baeumlein, Partik. p. 278), is not merely verbally correct, but also in keeping with the character of a μυστήριον; while, according to the usual way of taking it, the first half at least contains nothing at all mysterious, but something superfluous and self-evident.
George Benedikt Winer wrote,4


The Implication
Because the apostle Paul wrote «πάντες μὲν οὐ κοιμηθησόμεθα πάντες δὲ ἀλλαγησόμεθα», that is, "no one shall sleep, but all shall be changed," he believed --- at the time he authored his epistle --- that he and those Christians whom he addressed would experience the advent of the Lord Jesus Christ before they died. Of course, it was at a later time that the apostle Paul was resigned to experiencing death (cp. 2 Tim. 4:6-8) and at some time thereafter experiencing the resurrection of the dead instead of being changed while still alive and not experiencing death.
Footnotes
1 p. 493, πᾶς, III.
2 p. 460, οὐ, 2.
3 p. 384-386, 1 Cor. 15:51
4 p. 555-556
References
Meyer, Heinrich August Wilhelm. Critical and Exegetical Handbook to the New Testament. Trans. Moore, John C.; Dickson, William P. New York: Funk, 1889.
Thayer, Joseph Henry. A Greek-English Lexicon of the New Testament: Being Grimm Wilke’s Clavis Novi Testamenti. Rev. ed. New York: American Book, 1889.
Winer, George Benedikt. A Grammar of the Idiom of the New Testament. 7th ed. Andover: Draper, 1892.