Let's continue the digging on this. Another important fact we have to bear in mind is that lockdown is not digital. It's not ON/OFF. It's a continuum of measures (social distancing, school closures, social events, etc.) Same, relax of lockdown is not ON/OFF, it's also gradual.
So if want to see how it changes R, we need to quantify the lockdowns. Luckily, Oxford econ has you covered as they have a team of 100 monitoring lockdowns over the world and creating a "stringency index" which quantifies the measures taken to reduce contacts (so reduce R).
Rather than some absurd R change over 1 week, let's look graphically (often better than shitty statistical analysis) at how R and "Lockdown Stringency index" moved over time!
The UK (just a tiny relaxation)(All charts: R is black, Stringency is red) 
Spain : relaxation, spike in R, now under control
Austria: big relaxation, substantial increase in R
Canada : tiny relaxation, small increase in R
Germany: relaxation, increase in R
France, relaxation, R barely moves
Sorry, forgot the chart
Japan: OMG it's complicated, maybe the Olympics changed the numbers ;-)
Korea, super interesting: they relaxed big time, small increase, then a big cluster which they managed to get under control
the US, tiny relaxation, R moves are just tiny noise
And the Chinese ! Extremely interesting: a cluster, R spikes and then BIM new lockdown rules immediately implemented. During 1st relaxation they kept R under control.
So I guess this shows how more nuanced the real picture is - and R is generally going up after easing, but controlling it is the key!
