The Harry Cox version did have the Rule Britannia chorus, thought the verse didn't use the verse tune. I posted words and music for that in Lyr Req: Death of Nelson, where you will also find a few other songs about his death. The song isn't about Trafalgar per se, but mainly about the aftermath and the memorial for Nelson (the song also uses the title Nelson's Monument).
You will find a couple of songs about Trafalgar in the Bodleian's broadside collection (Trafalgar Bay and Trafalgar's Battle (tune:Chapter of Kings) if my notes are correct), but neither has Rule Britannia in them. Did the song actually use the Rule Britannia chorus, or was the song sung to the tune?