I have looked through my records and found no recent Lawrence Tremeer. I did find a William as follows:-
1978,29-Mar,Tremeer William Thomas, Death, Caloundra Cemetery Queensland, Australi. There is a note that he was born 1892 Feb 2 but I can't find an entry for the birth. Is that possibly your grandfather?
I suspect that you are probably linked to the Bradworthy Tremeers as many of them from that part of North Devon seem to have emigrated in the mid 1800's to Canada or Australia. My lot, from South Devon seem to have stayed put in Devon.
There were a few Tremeers in and around Dover, some in London and others scattered around. Cornwall is presumed to be the origin of the name and I have records [not linked, just a list of names, going back to around 1400]. If you can post some more detailed information about your ancestors including their names, dates and places of birth when the emigrated etc then others on the list may be able to pick them out and attach them to an existing tree.
If you want more information reply directly to me at firstname.lastname@example.org