I buy most of my non season tickets via the GWR website and collect from TVMs, my default station is Taplow but I've never had any problems collecting them from TVMs at other stations, however despite using the same card to pay for/collect them I'm always asked to input the code.
I buy most of my tickets using the GWR website or app and usually collect at a Southern or SWT station. I always have to input the collection code. Using SWT website and collecting at SWT station is the same. However booking from Southern website and collecting from Southern station I did not have to input code. I've not done that for a while so don't know if that's still the case.