I'm looking for the population and GDP of the UK nations (England, Scotland, Wales and Northern Ireland) from the 1950s to today, ideally at the yearly level. Any tip?
The Office for National Statistics website is going to be your friend here.
The current Population Estimates Dataset includes (under 'Mid-2016 detailed time series') national and regional population estimates going back to 1838.
For GDP there are several options, so it's probably best to look through the available data by searching the ONS website for 'region GDP' (note this link restricts the search to times series). One option is the Historical Regional GDP dataset, but that only goes back to 1968.
If you can't find exactly the data you need, you can also ask ONS to release additional data.