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He was born in Switzerland around 1803 and had resided in the UK from around 1828 when his daughter Margaret was born.
He entered a mark at Goldsmiths Hall as a watch case maker from 4 Richmond Buildings in 1836 and was still there for the 1841 UK Census living with his English wife Elizabeth and their six children also born in England.
The Directory of Gold & Silversmiths Jewellers & Allied Traders 1838-1914 Vol. I p. 367 by John Culme notes that as Elizee Piguet he entered a maker’s mark in 1840 with Philibert Matthey, in 1841 with Armond Correvon and later in the same year another mark alone.
His son Charles entered his own mark from Richmond Buildings in 1847
They were living at Hollen Street, Westminster in 1851 and at Fisher Street, Finsbury in 1861 where he continued as a watch case maker.
No further UK records have been found for him.

Further detail of Philibert Matthey can be found here:

http://www.925-1000.com/a_OB_Mathey1826.html

The likely reason Abram's son, Charles, entered his own mark:

THE COURT FOR RELIEF OF INSOLVENT DEBTORS.
Saturday the 17th day of April 1847.
ORDERS have been made, vesting in the Provisional Assignee the Estates and Effects of the following Persons:
On their own Petitions.

Abram Elizee Piguet, late of No. 4, Richmond-buildings, Dean-street, Soho, Middlesex, Watch Case Maker.—In the Debtors' Prison for London and Middlesex.

Source: The London Gazette - 20th April 1847