Why not use the ONS UPRN Lookup files?

[SK53]

The ONS lookup files, which Owen Boswara has verified are open data, have a lookup for each UPRN to postcode. These might be a better starting point than the raw Open UPRNs.

The assignments are not always correct because postcodes are assigned to things like post boxes, street records (IIRC), substations, ponds etc which do not have postal addresses, but given that these exist in the main file it's probably best to filter out those UPRNs anyway (e.g., uprns not in OS Local buildings layer).