Not locating the family yet in the 1851 census although there is an 1850 record (and earlier) for an Andrew Dinnen in the Hampshire, Portsmouth Electoral Registers 1835-1873 collection.
ADDED: Not sure if I have the correct Dinnen family, but the children below look interesting. The 1841 census does include Caroline, Ann, and Elizabeth.
1859 marriage - Elizabeth Aldina Revous Jackson (nee Dinnen) to George Robert Symonds.
Friday 08 July 1859
Shrewsbury Chronicle
"G.R. Symonds, Esq., late of the Ordnance Department, Malta, to E. A. R. Jackson, second daughter of the late Andrew Dinnen, Esq., of the Naval Dockyard. Portsmouth, and sister of John Dinnen, Esq., Admiralty, Whitehall ..."
Jan 1866 - Death of an Inspector: Mr John Dinnen (age 59) an inspecting officer of the steam branch of the Royal Navy, when leaving Whitehall, was knocked down by a Hansom cab..." (lots of newspaper deaths notices for this accident)
Saturday 30 March 1867
Naval & Military Gazette and Weekly Chronicle of the United Service
"Miss Anne Dinnen and Miss Caroline Dinnen, sisters of Mr John Dinnen, Inspector of Steam Machinery Afloat, who lost his life by being run over in the street when ..."
Friday 17 June 1910
London and China Express
"Miss Dinnen died the 22nd of December, 1909; she was the daughter of Andrew Dinnen, sometime of Portsmouth, England, and Isabella Winstanley, his wife; these left several other children...."
The death notice above is for Caroline Augusta Dinnen, born 1820. C.A. Dinnen (age 81) is a patient in the 1901 census in Camberwell.
I know the details above don't answer the OP's question re Andrew's birth record, but just wanted to mention them in case they help with tracing the family in 1851.
