I found an example of "Water Water Wallflower" in an old book of African American play songs. I posted it in the Mudcat thread "Children's Street Songs" in response to this query thread.cfm?threadid=4300#1372419

Here's that example and my comments:

Subject: RE: Children's Street Songs
From: Azizi - PM
Date: 05 Jan 05 - 06:35 PM

...

In "Play Songs of The Deep South" Altona Trent Johns, 1944
tthere's a version called "Water-Flower" The first verse is completely different:

1st verse: Water-flower, water-flower
          growing up so tall
          All the young ladies must surely, surely die
          All except 'Lindy Watkins,
          She is everywhere=
          The white folks say, the white folks say,
          Turn your back and tell your beau's name.

end of quote--
But the second verse of that song has the verse you quoted:

2nd verse Doctor, doctor, can you tell
          What will make poor 'Lindy well?
          She is sick and 'bout to die
          That will make poor Johnnie cry.

end of quote

As you can see that's the same as yours except for the change in names. Now if your name is Ronnie and your sister's name is Nancy that would account for the name changes...

"Water-Flower" has a different ending than the one you gave:

3rd verse Marry marry, marry quick!
          'Lindy, you are just love sick!

4th verse Johnnie is a ver' nice man,
          Comes to the door with hat in hand,
          Pulls off his gloves and shows his rings,
          "Morrow is the wedding-day.
---
"Water-flower" is described as a pantomine ring {cicle} game with one girl in the middle; a boy is said to act out the role of the doctor {This was before Women's Lib} and the doctor selects the boy whose name 'Lindy had mentioned to come into the center of the ring and act out the role of "Johnnie"..."