Probably because he is PM of UK and felt the need to communicate to 30 million households to emphasise the importance of the stay at home message. He mentioned each home nation and their NHS staff in his letter.
He did not do it because his Government is in charge of UK healthcare because its not.....it’s devolved, that’s a fact.
I hope he got the approval of the first ministers of each country, but he is pig headed enough to go it alone. In Scotland households got two letters. Nicola Sturgeon sent her own at the same time, possibly to emphasise that she’s in charge of Public health and to take the wind out of his sails