I think that you are in for a long hard slog, I'm afraid.
You will probably need to assume that there is a connection between John & Thomas. Then you will need to list possible combinations (e.g. Thomas Corkel, son of Matthew and Ann Skealee, was christened in K. German on December 12 1794. Thomas Corkill and Isabel Christian's children included John Corkill christened July 14 1755 and Matthew March 3 1761).
Then you need to to validate or eliminate the combinations (in the above example, getting John's birth year would help. Do you know when he died? Was it after the 1841 census?)
I notice that there were lots of Corkell's in and around Penzance. Are these your people?
Ann was married in the December quarter of 1837 volume 4 page 286. Her marriage certificate might indicate if her father was still alive at the time. What is strange is that her marriage is noted as Penzance while all the other people on the page were married in Lambeth. Probably a mistake - place of birth may have been transcribed as place of marriage.

I do not know about the connection with Kevin's Corkills, but in any given census, there were about 500 with the name on the island and another 100 or so in England. That is before adding in the Corkells and Corkhills. I had assumed that the carpentry connecion was through his future brother-in-law Barnicoat.

Also try this site on October 8 1834, where is a burial for a William Corkill in Penzance 9 months.

http://west-penwith.org.uk/pzbur2.htm

The parish records may have some information on his parents.
http://west-penwith.org.uk/pzbur2.htm

Regarding coast guard records
There is information on coastguards at:
http://www.mariners-l.co.uk/UKCoastguards.html
It does not look very promising, but if you have time, who knows?