What's been so interesting here in the States is that, while Trump has been incompetent at best and actively dangerous at worse (and gross throughout), we have a federated system.
Most of the big breakthroughs you're seeing don't come from the federal government but from state and local government, as well as from private enterprise. That's why you'll see some of this happening piecemeal.
California, NY, and Washington are going to go strongest soonest, partly because they're populous and partly because they're liberal. Right now there's a 30-40 point difference in how people plan to react based on political party, not least because both the president and, even longer, Fox news said that this was a liberal hoax. (That's the dangerous part.)
But you see this with the universitieseach university decided on its own, and then they all started to. And you see this with school districtsin some states, adjacent school districts are doing different things. The thing is that once one place starts to do it, the others often soon follow. It's a lemming-cascade effect.
On the other hand, the central government has resources that states really don't, particularly the poorer ones. The legislation that just got passed, for instance, could make a huge difference and is only possible at a national level, but it's taking a long time because it's caught up in some incredibly dumb politicking, particularly from the senate and early at the White House. (I never thought I'd be so delighted with Steve Mnuchin than I have been in the past five days--Trump refuses to meet with Pelosi because he thinks she'll humiliate him.)
The UK is significantly and enormously more centralized than the States. Sometimes this is a good thing. But other times, like when central government seems paralyzed, it means it lacks the ability to change nearly as quickly.
TL;DR: the safety measures we have in the States by and large don't come from Trump but from governors and even mayors.