There was no formal system of adoption in England in the eighteenth century. In fact, it was not until the Adoption of Infants Act of 1926 that there was any centralised register for adoptions. Prior to this date, most adoptions were informal arrangements.
In your case, there were two likely outcomes. If there was family money to raise the children, there was likely documentation of that, in the form of a will or administration. In this case, there would have been a paper trail since the infants were, of course, unable to administer their parents estate and would likely not have access to the money during their minority. In such a case, a guardian and trustee might be appointed during the probate process. Since a gravestone was made – not only that, but one that has survived to the present day – there was clearly enough funds in family to afford a gravestone, but that is not to say that the infants had money left in trust to them. It is definitely worth ruling out if there are any probate records that could help solve your problem. The starting place is the Nottinghamshire Archives catalogue. Most Bingham wills would have been proved in the Archdeaconry of Nottingham (and Deanery of Bingham), so you can search using the reference PR/NW. Also to be sure check the higher court (Prerogative Court of Canterbury) in case your ancestor was wealthy or had assets in another jurisdiction.
It is equally likely that there was no money to raise the orphans, in which case the parish would have been responsible for providing for them. In many cases orphans would be raised by family or friends, and may have been eligible for parish relief. There again should be documentation of any parish relief, however whether that documentation has survived is not a given. Typically my starting place would be poor law records such as churchwardens' accounts. These records document the money that was paid to the poor. It is not uncommon to find entries that state something along the lines of "Paid to John Smith for Jones children", which would suggest that John Smith was providing room and board for the children with payment from the parish for their upkeep. Such situations might not have been permanent, and the children may be separated or moved around as the parish saw most expedient. For Bingham, again I would start at the Nottinghamshire Archives. I cannot see any churchwarden's accounts for Bingham in the catalogue, but the catalogue is not complete. Checking on FamilySearch shows film 1526119 contains Parish chest material for Bingham (and luckily the film is digitised and accessible from home). The film contains churchwarden's accounts starting in 1769, which is a little late in your case, but worth checking in case the parish was still providing for the children at that point.
It also should be considered whether the children were placed within a public or private charitable institution. There were few workhouses and orphanages outside the large cities in the mid-eighteenth century, but it cannot be ruled out completely.
All this being said, I think it first is imperative that you determine whether the children were actually orphaned. I viewed the source you stated that shows the father died in 1765 as well, but it is unsourced, and I can find no burial entry to support that. Before going down the wrong road too far, I would inquire with the Bingham local history society to ask if they could provide their source, or put you in touch with the person who compiled the tree. It might just be a mistake, in which case the children might not have been orphaned at all.
