When viruses mutate and the surface of a protein changes that usually results in the antibodies that the body produced against an earlier form of the virus working less well. By what factor do antibodies developed against the Alpha-SARS-CoV-2 spike protein bind less well to the Delta-SARS-CoV-2 spike protein?
There is no general answer to this question -- it depends on the viruses involved, the antibodies in question and the specific immune system gene versions of an infectee.
Unlike with monoclonal antibodies, the population of antibodies to a given protein generated by a natural immune response is enormous. That population includes antibodies targeted to different parts of the protein, as the immune system chops up the protein into short (8-20 aa long) peptides in order to "present" those to the various antigen-recognizing immune cells. "Peptide length significantly influences in vitro affinity for MHC class II molecules" doi: 10.1186/1745-7580-4-6
Any given change to the original target protein may directly or indirectly affect from just one to many of its recognized peptides. Thus, a given antibody may have an affinity for the new protein ranging from unchanged (if its target antigen region was unchanged) all the way to essentially no binding (if its target region was greatly changed or eliminated). An individual's overall Ab "neutralizing titer" against the new protein will depend both on the affinity of individual Abs to the new protein and on the number of each Ab present.
In vitro research suggests that the multiple changes associated with the S- (Spike) protein and for that matter, the N- (nucleocapsid) protein attributed to omicron that has demonstrated some ability to evade current vaccine-induced immunity, but also, and more markedly, can evade infection-conferred immunity. Or, to put it another way, recent infection with the delta variant is little protection against a breakthrough infection with omicron. Recent prior infection appears to moderate the potential severity in some unvaccinated, but previously infected patients. Persons who have been fully vaccinated and boosted may also experience breakthrough infection but the majority are mild symptomatically,with a much lower predisposition toward requiring hospitalization. Persons previously infected and subsequently fully vaccinated, or fully vaccinated but who still experienced breakthrough infection are much less likely to contract a breakthrough infection and much more likely, if they are reinfected, to have asymptomatic or very mild disease.
I'm not sure I've seen any research, peer-reviewed or preprint, that discusses the IgG binding affinity for the case you describe.