You're going to need to do some "reverse geocoding" to make this work. I don't think that can be done directly in Tableau and would probably require that you write a bit of code (might also be possible with a tool such as Alteryx). Is that an option for you?
Where are you located? If you are in the US you can go to the Tiger/US Census site and download the data - it only comes in National level so it is lot of data
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This is where Python is your friend.
Look up geocoding with python.
If you're not familiar with python, I suggest start by installing Anaconda, and use Jupyter Notebook for your interface. Plenty of docs all over to help get ya rolling.
Google's Maps api use to be able to encode a lot of Lat Longs, zips, addresses etc... but I think it stopped being free in July 2018. Not sure what the go-to geocoding script is right now, but this one use to work well. https://gist.github.com/shanealynn/033c8a3cacdba8ce03cbe116225ced31
After you get your zips, just join them into your data off the lat and longs.
Much depends on the specific location for your postcode data. Where are these postcodes located?
For instance in the United States a zip code is not really a geographic entity. You can't back-calculate your way into a reliable zip code from lat/long values.
Other jurisdictions have different rules.
Thank you all for your help.
My postcodes are mainly located in UK / Europe.
I will try to go for the "reverse geocoding" option, even if I am not very comfortable with it at all. I don't have access to Alteryx, but I will investigate if that could also work for my purposes.