本人在香港出世及曾擁有兒童身份證, 由於小時跟父母移民到英國, 现兩文件已遺失 现慾補领成人身份證, 不知有何方法 請賜教?

You can apply for a replacement ID.

Given that you have a child ID, the Immigration Department should have your fingerprint on record. The Immigration Department can verify your identity with your fingerprint and your British Passport.

It is best if you can prepare a police report for the lost ID.

Although a birth certificate is not a must for the replacement, in case you want to obtain one, see here:

http://www.immd.gov.hk/hkt/services/birth-death/Application_for_Search_of_Birth_Records_and_Certified_Copy_of_Birth_Entry.html

In your case, a record search must be done before a birth certificate can be issued.