I have done a good deal of Gardiner research since I posted that message, and I thought I would update it.
Using film #1526731, Parish Records of Somerton, Somerset County, available through LDS Family History Center, I learned the following:
Almost certainly, James Gardiner's mother and father were William and Unis Waldron Gardiner, who were married in Somerton, Somerset County, England, in 1805.Unis's name is variously spelled, in the records, as Unis, Unity, Eunice, and Unice, but I decided on Unis, as that is the name on her marriage record and on her burial record.
James's date of baptism was 7/16/1825.A note next to his name says that he was born in March 1820.
Siblings of James Gardiner seem to be;
(1)Elizabeth Gardiner, baptized 4/10/1817, who possibly was buried 2/12/1840 at age 21.
(2)William Gardiner, baptized 2/20/1824, who probably died as an infant and was buried 12/20/1824.
(3)Benjamin Gardiner, who was baptized 7/5/1825, and who married Mary Rays in 1847.
(4)Mary Gardiner, who was baptized 5/8/1817.A note next to her name says that she was born at Thornton Marnhill, Dorset County, in March 1815.
James and Caroline Gardiner had 11 children that I can account for, all christened in Somerton.They were:
Charles Gardiner, bap. 3/2/1844.
Edith Gardiner, bap. 10/12/1845.
John Gardiner, bap. 8/2/1847.
George Adam Gardiner, bap. 9/11/1850.
William Westlake Gardiner, bap. 1/2/1853.Born 11/30/1852.
Jane Gardiner, bap. 1/3/1855.
Henry Gardiner, bap. 4/24/1856.
Harriet Gardiner, bap. 12/14/1858.
Edwin Gardiner, bap. 6/7/1861.
Robert Gardiner, bap. 10/21/1863.
Emily Ann Gardiner, bap. 6/3/1868.
I have some more information, and am in search of still more.I hope this helps somesbody who is researching this family.