The Family History Library did not have the resources available for me to find Paul Doughty's enlistment regiment. I was able to contact a researcher in England who does research at The National Archives. I gave her the information I had about Paul Doughty. She was able to search for Paul Doughty in the London Gazette. According to the The London Gazette's website it is the "Official Newspaper of Record for the United Kingdom" and each issue has a Ministry of Defence Supplement that includes "officer’s appointments, promotions and similar within the Services" (http://www.london-gazette.co.uk/). The Gazette began publishing in 1665.
The following information was extracted from the Gazette:
18th January 1780: Sergeant Doughty of 1st Foot Guards to be Ensign in an Independent company of invalids at Portsmouth.
15th April 1794: Ensign Paul Doughty to be Lieutenant in Captain Campbell’s Independent Company of Invalids.
Finally, here was Paul's original regiment: the 1st Foot Guards.