They do save some messages but it would be interesting to discover how much they have access to and whether security protocols are followed (I bet they aren't) -
Earlier this year, a witness statement was given to the High Court by Sarah Harrison, the chief operating officer for the Cabinet Office, in relation to a court case brought by public interest group The Citizens – challenging the Government on its saving of WhatsApp messages. In it, Harrison claimed that “various WhatsApp groups had been set up [by those in Downing Street]. These were largely used for general discussion or informal conversation”.
“Anything relevant to public record would be saved either through the Prime Minister’s Private Office support team (who provide 24 hour administrative assistance),” said Harrison, or “by actions being formally commissioned by officials through Government channels and/or the box process, in accordance with the Number 10 WhatsApp policy.”
https://bylinetimes.com/2022/04/27/government-stalling-on-releasing-boris-johnson-whatsapp-messages/