Apprenticeship during the 19th century was usually a private arrangement between families. After 1811 formal records were not required for taxation purposes. So any documentation found would generally be a case of chance survival.
But there are two main exceptions for London. One would be if the apprentice was working within (or very close to) the City of London boundary and his master was a member of one of the City livery companies. In that event the apprenticeship may be referred to in the relevant Company's archives. These are mostly at Guildhall Library, although some companies have retained their own records. (The apprenticeship records for London livery companies that have been indexed online, e.g. at FindMyPast and Londonroll.org, are mostly pre-1800.)
Another exception would be if the family was very poor, when an apprenticeship might have been paid for by the poor law guardians or by a local charity. In that event there might be something at the record office (LMA), or at the local studies archive for the relevant London borough.
But unless you have a positive lead to either of the above circumstances, I'm afraid it probably isn't a practical search.
