These extracts are quite ancient for our use, but may help throw more light on the subject of mixed bathing, and especially with regard to the use of nosegays!

The first from Samuel Gale A Tour Through Sev’ral Parts of England (1705)

The situation of the baths is promiscuous, in several of the street of the city, and surrounded with high buildings, from whence spectators from the windows may view the company when bathing, the surface of the water being entirely open to the heavens….

The manner of going in is for the gentlemen and ladies to dress themselves in their proper habits in their own apartments: the first in fine canvas waistcoats of a sandy colour edged and trimmed with black ribbands or ferreting, and tied down before with strings of the same colour, having on canvas drawers and slippers and a lawn linen cap: the latter in canvas gown and petticoats with pieces of lead affixed at the bottom, to keep them down under the water. Being thus dressed they are brought in chairs,sometimes close covered up in their morning gowns, and are set down in the passages which lead into the bath, shut at each end for more privacy. The descent from the passage or entrance is by stone steps, at which one of the guides attending the bath meets you to conduct you in. The first we visited was the Cross Bath. Two sides of the bath have galleries, one for spectators the other for the music. This bath is the most frequented by the quality of both sexes, where, with the greatest order and decency,the gentlemen keep to one side of the bath and the ladies the other.No gentleman must whatever presume to bathe in the ladies’ district, under a pecuniary mulct, inflicted by the serjeants of the bath: the ladies are supposed to be so modest as not to come near the gentlemen…The ladies bring with them jappanned bowls or basons tied to their arms with ribbands, which swim upon the surface of the water ,and are to keep their handkerchiefs, nosegays, perfumes and spirits in case the exhalations of the water should be too prevalent. The usual compliment when any one goes into the bath, is too wish them a good bath: and the company while bathing, generally regale themselves with chocolate… 
Pp.21-22.

And this order from the Bath Corporation to tighten up the Rule on Bating Dress worn at the Baths.

It is ordered..by this Corporation that no Male Person above the age of Ten years shall at any time hereafter go into any Bath or Baths within this City by day or by night without a pair of Drawers and a Waistcoast on their bodies. And that no Female person shall…go into any Bath…without a decent Shift on their bodies. And that…each guide shall wear a Cap with a Tassel to it to distinguish them from other people.

Bath Council Minutes 26th September 1737.

It seems that the makers of NA took into account all 18th century sources when re-creating the bath scene.

The only thing they disregarded was the segregation of the sexes. But then I suppose John Thorpe would be prepared to be fined in order to impress the ladies with his merry banter.