The Online 78 rpm Discographical Project says that Herbert 'Happy' Lawson (the songwriter) recorded ANYTIME on July 31, 1925.

Emmett Miller, accompanied by his Georgia Crackers, made 3 recordings of ANYTIME in 1924, 1928, and 1936. (The Georgia Crackers, by the way, included Jimmy and Tommy Dorsey and Gene Krupa!) The Red Hot Jazz Archive has the latter two, from which I transcribed these lyrics:

VERSE: Now I'm so sad and blue,
'Bout nobody but you.
Why you don't say nothin' to me?
I told you that I loved you right from the start.
You told me the very same thing and now you try to break my little heart.
If you don't want me, why don't you come and tell me so?
I love you and I'll say just before I go:

CHORUS: Now, any time that you are lonely,
Any time that you are blue, oh, Lawd,
Now, any time you're feeling downhearted,
That will prove to you my love is true.
Now, any time you're thinking of me,
That's the time I'll think of you, oh, Lawd.
Now, any time you want me back again,
That's the time I'll come back to you, oh, Lawd.

Interestingly, Miller's singing style includes a 'blue yodel' that is very similar to that of Jimmie Rodgers.

The Red Hot Jazz Archive also has a recording of ANY TIME by Clarence Williams' Jazz Kings from 1928. Williams has messed around with the lyrics quite a bit, and doesn't even credit Lawson as the songwriter, but you can hear strong similarities:

Any time, honey mine, you feel lonely,
Any time, honey mine, you feel sad,
Oh, any time, baby mine, you need leg (?), dear,
Send for me and 'twill be just too bad.
Any time, honey mine, that you need me,
Take your phone, call my home, one-eight-nine,
'Cause I've got just what it takes to make you happy,
Any time, honey mine, any time.

The Lester S. Levy Collection of Sheet Music has the sheet music. It gives 1921 as the copyright date, although this particular printing of the sheet music is obviously from much later. Also, it omits the verse that Emmett Miller sang.

"The Reader's Digest Country and Western Songbook" also has the sheet music of ANY TIME, and it seems to match the Levy copy. It says Eddy Arnold recorded it in 1948, Eddie Fisher in 1951, and Patsy Cline in 1963, although her recording wasn't released until 6 years later.