An interesting enquiry.
I have checked the National Archives database which has the records of those that fought at Trafalgar and which is based on the information recorded in the ships muster books. Two John Pringles are listed:
John Pringle Able seaman age 24 born in N.York served on HMS Euryalus
John Pringle Volunteer 1st Class age 14 born in Dunbar Scotland served on HMS Defiance
The Forces War Records database also has these two men listed on the Trafalgar Medal Roll.
So no confirmation of your coxswain John Pringle with Nelson on the Victory at Trafalgar.
However a Google search brought up a site
http://www.jamespringle.co.uk which has pages from a book called The Records of the Pringles (a book that appears to have been published in 1933). In this it mentions a John Pringle ( page 328), born in Fife, who joined the Royal Navy when he was 21 and in 1784 became coxswain to Nelson who then commanded the frigate HMS Boreas in the West Indies.
It goes on to say that he took part in other celebrated naval battles including Trafalgar and the Nile but unfortunately doesn't give any hard information as to the ships he served on, or if he still served with Nelson at all after his West Indies posting.
So there appears to be some truth in the story but like many of these tales, the facts and timelines get easily blurred.
I hope this information is helpful and good luck with your future research.
Phil