The Foundling Hospital records at the LMA are an amazing collection of documents.
I've looked at foundling records for people on here
http://www.rootschat.com/forum/index.php?topic=393923.0http://www.rootschat.com/forum/index.php?topic=600056.0It is time consuming research but if you have their baptism date and their entry number, you can check the documents held.
These usually contain the application made by the mother for admission to the hospital, not all applications were successful. The applications were heartfelt, explained the mothers personal circumstances and were generally seconded by another person as to why the child should be accepted.
You will find out their original birth names and sometimes the names of the putative fathers.
You may be able to see where they were sent to be be looked after as babies.
You may also find out where they served apprenticeships or were sent as domestic servants.
You can ask the LMA to undertake this research for you but as it is extensive it will be expensive, £80 per hour.
I would volunteer but unfortunately I now work during the week and cannot commit, sorry.