find tables and apply table-style to all

The name of the table style should be changed in the script;

style = app.documents[0].tableStyles.item('My_Table');
tables = app.documents[0].stories.everyItem().tables.everyItem().getElements();
for (i = 0; i < tables.length; i++) {
  if (tables[i].storyOffset.appliedParagraphStyle.name == 'TableText') {
    tables[i].appliedTableStyle = style;
  }
}

Sorry, Caroline, I should have read your post more carefully. The tables you're after are in a paragraph style TableText, and you want to apply a table style. You didn't say what the name of the table style is, in the updated script, below, I assumed TableText (same name as the paragraph style). If you use a different name, replace TableText in the script's first line with the name you use.

style = app.documents[0].tableStyles.item('TableText');
tables = app.documents[0].stories.everyItem().tables.everyItem().getElements();
for (i = 0; i < tables.length; i++) {
  if (tables[i].storyOffset.appliedParagraphStyle.name == 'TableText') {
    tables[i].appliedTableStyle = style;
  }
}

It'll have to be a script:

style = app.documents[0].tableStyles.item('TableText');
tables = app.documents[0].stories.everyItem().tables.everyItem().getElements();
for (i = 0; i < tables.length; i++) {
  tables[i].appliedTableStyle = style;
}

P