If it does occur during meiosis, are the alleles switched over from one sister chromatid to the other during the crossing over process in Prophase I?
Crossing over/synapsis in Prophase I of Meiosis I is when alleles are switched over from one non-sister homologous chromatid to another non-sister homologous chromatid. This happens before fertilization ever happens. Meiosis occurs even before birth in females in order to make gametes in the form of oocytes. In contrast, this frequently occurs in men who continue to produce gametes in the form of spermatocytes throughout their lifetime. Fertilization simply refers to the fusion of two gametes-- a spermatocyte and oocyte combining together-- and does not involve meiosis.
Here's a short and straightforward Youtube video about Meiosis.