Reading through your post, I'm not sure what evidence there is for saying he was born in Launceston.
Do you have something prior to his marriage in London that places him somewhere other than Greenwich, which at this time was classed as Kent.
Have you seen the transcription on findmypast of Joseph Bounsall taking on an apprentice in 1786, where he is given as a master tailor of Plymouth? (Comes under the 'Education and Work' heading).
This transcription is from The Society of Genealogists , and held at The National Archives at Kew. It gives all the reference numbers to be able to get hold of it should you want to.
There is also a similar apprenticeship record in 1804 for a 'Josh' Bounsall at Maker, which is a place about 9 miles from Plymouth but just in Cornwall. He was a Cordwainer (shoemaker).
And also one for a James Bounsall in 1785, also a master tailor, also at Maker. (I wonder if this is your Joseph's son? Born in 1762, he would be 23 years old by 1785).
This would suggest that to become a 'master tailor', Joseph himself would have had an apprenticeship. Perhaps this was why he was in London?
There is a website 'londonroll', which has records of thousands of apprenticeship records.
https://www.londonroll.org/I had a quick look, but there does not appear to be any soundex for variations of name, so you have to put in all the different possibilities of Bounsall. I couldn't see Joseph but you may like to have a more thorough look. Don't worry about trying to pick a particular company he might have been in - I have found you didn't necessarily get an apprenticeship in the company of your trade. For example I have a dyer who was apprenticed in the Drapers' Company, and also workers on boats on the Thames who were also apprenticed to the Drapers' Company. Just pick 'all companies'.
Apprenticeships were usually for 7 years, more often than not started when the apprentice was about 14, and one of the conditions was that apprentices should not marry within that time.
(If James the master cordwainer in 1785 IS Joseph's son, there should be an apprenticeship record for him too somewhere.)
As your Joseph was married in 1761, one supposes he was possibly about 22 years old by then, so born about 1739.
As you say, there do not seem to be any births on the usual sites.
There is an expectation that ancestry, findmypast and familysearch have all the records that exist.
This is of course not true, and sometimes we have to find other avenues to search.
There is a website, the Cornwall On-line Parish Clerk.
https://www.cornwall-opc-database.org/search-database/There are many listings for the Bounsall name (plus variations).
A Joseph Bounsal married Ann Ferris in 1720 at Launceston St Mary Magdelen.
They had several children baptised at the same church.
On 30 August 1721, a Joseph Bounsall was baptised, father Joseph, a weaver.
This child was then buried 13 August 1722.
Baptisms of several other children until another Joseph Bounsal, baptised 5 Nov 1737, father 'Jos:', a weaver.
There is no corresponding burial for him as a child.
familysearch.org has the films of the Bishop's Transcripts which is viewable from home if you're signed in. They don't seem to have the ordinary registers for St Mary Magdelene's church, which is a bit odd. Perhaps none survive.
(Don't search 'records' on familysearch, but go to 'catalog', and put in the place name Launceston. Go down to 'church records', and then find the right film.)
Unless you know something else about him being from Cornwall, it's still not proof that this is the Joseph Bounsall you're looking for, though it looks a possibility.
Normally, the next course of action would be to search wills to see if you can join up the dots.
Unfortunately, most Devon (and Cornwall), wills were destroyed during the bombing of Exeter in WWII, as they were housed at the cathedral there.
There is a site of The Consolidated Wills Index for Devon and Cornwall
https://www.cornwall-opc-database.org/search-database/There are some Bounsall entries, but nothing that seems the right time for your Bounsalls. Plus many of these entries tell you there USED to be a will, but it is now missing, so I always find it a very frustrating site!
It's worth looking at though for information.
There ARE a few Bounsall wills on the Cornwall archives site
https://kresenkernow.org/.
Can't see any that fit with yours, but you never know.
Have you searched this Cornwall site, or the Devon one at
https://swheritage.org.uk/devon-archives/for any information that might be held there about the family?
