Interestingly, Scotland now has 'baby boxes' too [0] - but not the rest of the UK, it is an initiative by the devolved Scottish government only it seems.
The box itself is great, but the reason that Finnish mortality rate for babies fell so sharply after the box was introduced was because to qualify for one you had to visit a doctor early in your pregnancy.
Having early contact with healthcare, when pregnant, means you're more likely to do well. The doctors/nurses/midwifes/assistants can frown at you, tell you to eat your vitamins, stop smoking/drinking, etc.
For what it is worth healthcare is not free in Finland, but we paid about €300 for mother, father, and baby to stay in a private room for 3-5 days. (Just over a year ago, I can no longer remember that period. I just remember in the 3-5 days I drank a full bottle of whisky, and smoked a cigar in the car-park, in subzero temperatures. Finland, remember?)
Interestingly, Scotland now has 'baby boxes' too [0] - but not the rest of the UK, it is an initiative by the devolved Scottish government only it seems.
0. https://www.mygov.scot/baby-box/