Hi, I've been looking too
As grandarog doubtless found, this seems to be the only possible birth record for a Lilian Mildred Smith.
The baptism record says that Thomas Henry is an accountant (I think) but I also was wondering if he was/had been on the Navy because there is just one Marriage record which seems to fit a Thomas Henry Smith and Florence Amelia and that is June qtr 1881, Plymouth, vol 5b, page 500. and her maiden name is Hexter.
There are also some Naval Records on The National Archives site for Thomas Henry Smith...several in fact.....so I don't know which, if any, would be for him.
Unfortunately, they doesn't seem to be a birth or census records for Florence Amelia Hexter. Or for Thomas Henry Smith and his family beyond the marriage and baptism but I will keep searching
P.S. Maybe that isn't the birth record, as after searching for a while I can find possible censuses (including the 1911) which seem to muddy the waters. Lilian Mildred does seem to have been born around 1896/99 going by her London Marriage Record, possible 1911 and 1901 censuses, so a birth record of 1889 would be for someone else. Also Thomas Henry says he's an accountant but I may have him on one census as a coastguard.
Don't think I'm helping a lot here
Perhaps I'd best play quietly on my own in the corner