I'm traveling to London for what will be about a 5 month trip. I'm a UK citizen (by descent), and have never lived in the UK. With me will be my son, a US citizen (and not UK), who is 4 years old. Will it be possible to enroll him in (preferably a public) school? Alternatively, what other kinds of non-formal educational environments could I look into? (I.e. day programs that an adult can accompany a child to). My apologies if this is not the right Stack Exchange site for this question. I realize it might not qualify as an expatriate-related question, but I figured many expatriates might have done a similar process to the one I mention. Any help or tips are greatly appreciated.
From the information you have provided, you will not be able to enroll him in state education (cf https://www.gov.uk/guidance/schools-admissions-applications-from-overseas-children). UK schools start recpetion, something like a cross between pre-k and kindergarten, has a cut off of turning 4 by 1 september, so depending on your son's birthday he may not be eligible no matter his nationality.
Many parents in the UK do not send their children to reception, but rather keep them in nursery. For example, when my son left nursery for reception about 1/4 of his nursery class stayed behind and skipped reception. For a 5 month visit, starting late into the year, this might be a better option than a fee-based school.