It was part of the Wirral Rural District from 1894 until 1933. In 1933 Wirral Rural District was abolished and Caldy became part of the new Hoylake Urban District. [There is conflicting evidence here. Both GENUKI and A Vision of Britain through Time state that Caldy was transferred to Hoylake Urban District in 1933, but Wikipedia says Wirral Urban District.]
The population of West Kirby was 92 in 1801, 142 in 1851, 202 in 1901, and 607 in 1951. At the time of the 2001 Census, Caldy had 1,290 inhabitants.
In the nationwide municipal reorganization of 1974 the entire Wirral Peninsula, with the exception of the municipal borough of Ellesmere Port and urban district of Neston at the southern end, became the Metropolitan Borough of the Wirral in the new administrative county of Merseyside.
For more information, see the EN Wikipedia article Caldy.