I don't see why deaths are on course to double every few days now, Spain and Italy have never topped 1000 per day, I see no reason why we will.
It depends on when the lockdown was implemented really. UK deaths are doubling every 3-4 days. Just 7 days ago we were at 150-200 deaths a day. Italy locked down on the 8th March - it took around 14 days for the deaths to stop increasing and plateau at around 800 a day. Which makes sense given the incubation period for the disease is around 2 to 12 days, and deaths occur around 21 days after infection on average, so there'd be some lag from initially imposing a lockdown to seeing those affects come through.
The UK locked down on the 24th - 14 days after would be the 7th of April - so all things going well we should see deaths plateau in rises early next week. Without the lockdown, like basically the last two weeks, we'd probably see deaths continue to increase every 3-4 days, perhaps even faster given the NHS would start to get overwhelmed.
How long before deaths start to actually go down is another question - probably around 20-25 days after the lockdown given that's the average time for death from infection, and new infections should decrease a lot from the start of the lockdown.
A bit grim but hypothetically I wonder how many people would die if we didn't lockdown for another month? Assuming a doubling every 4 days, that would be around 8 doublings in 30 days, so the 600 deaths a day would go to 9600 deaths a day by the end of the month. Crazy.