Does a 1-year-old need any ID to fly within the UK?

It is VERY likely that the adult travelling with an infant WILL be asked for identity, AND to prove a parental relationship with the child. I'm sure you can work out why for yourself, considering the possibility for child abduction my estranged parents (or worse!).

I'm speaking from experience, as both myself and my partner travel alone with our youngster who is now 19 months old, and this applies both internally in the UK, and with external flights abroad (to a much greater extent!).

Policy DOES vary, and only the airline you intend to fly with can advise accurately, as rules change constantly... but arrive at the airport without ID/proof, and there is a good chance you will be refused access to board the flight.

ANY sense of something suspicious, for whatever reason, and you WILL end up talking to the police!