How I can find postcode from osm_id for UK?

I suspect that there won't be an already-existing complete database which contains a list of OSM IDs and postcodes. This is because many people who have added buildings to OSM haven't added their postcodes. For example, if you search OSM for your example postcode "EC1A 2DE" no exact match comes back. Nearby you can see this church (which has got a postcode mapped) and this adjacent building which hasn't.

It may be possible to infer a postcode for an OSM ID from various external sources - here's a blog post by @SK53 talking about one approach. I wouldn't underestimate the work required though.

Obviously postcode isn't unique except for large buildings - although maybe you're only interested in certain large buildings which do have a unique postcode?

I am afraid this approach is doomed to failure.

The problems are several:

• OSM Ids are system keys, they do not correspond to anything in the real world. They are not unique identifiers for things in the real world. A building may be represented by several osm objects, several buildings may be represented by a single osm objects, or a building may be built up from several objects in a relation.
• OSM Ids are not persistent, and may get reused in other objects. (The actual risk of this is small, as IDs are not reused once deleted).
• OSM Ids are only unique within a class of object (nodes, ways, relations). You need to use a compound key of object type and osm_id to ensure uniqueness within OSM.
• OSM has a tiny fraction (less than 2%) of all buildings in the UK mapped.
• Postcodes are not unique identifiers either.
• Postcodes are based on addresses (technically delivery points) not buildings: a building can have more than one address (e.g., a block of flats), many buildings share a postcode. Some buildings don't have postal addresses. Postal delivery offices may have hundreds of postcodes.
• Postcode geodata is provided with a centroid which should co-locate with only one of the buildings, there is typically around a 2% chance that it is the right one for houses, probably better for some commercial properties. For campus organisations such as universities the chance is much lower as all buildings share a postcode.
• There are no freely-available polygons which accurately show the area which contains locations for a given postcode. Using Voronoi polygons on the OpenData sets is not an adequate substitute.
• Postcodes are intermingled (a block of flats may have one postcode, and the buildings either side another different postcode). Therefore closeness to the centroid is only a partial indicator of matching.
• Postcodes are usually assigned on the basis of streets. It is possible for a centroid to be located closer to another street than the one it applies to.

If the data you have only contains the postcode as a building locator then it is not possible to find the building in any dataset. If you have some other criteria (for instance are the buildings public buildings) then this may help by reducing the candidates.

If you have addresses then it may be possible to enrich your data with a location by using Nominatim to provide a set of co-ordinates. Again this is chancy because OSM only has a small proportion of UK addresses mapped.

Sorry for having to be negative about your project, but I'd hate you expending a lot of effort for no result.

If you're prepared to do a bit of programming:

Download an extract of OpenStreetMap data for the UK from Geofabrik. Save this to a database, for example by using the osm2pgsql tool to save it to a Postgres database with the PostGIS extension.

Download the CodePoint Open dataset from Ordnance Survey. This is free of charge and licensed liberally. Add this to another table in the same database.

You can then create a spatial query to associate OSM IDs (in one table) with postcodes (in another). The PostGIS docs explain the functions you can use in writing your SQL.

This is not for the beginner, but if you have some experience in SQL databases, you shouldn't find it impossible.

Note that CodePoint Open only has the centrepoint of each postcode area, not the full boundary; so this approach will sometimes suggest postcodes which are slightly 'off'. The only way to avoid this is to buy the expensive full data from Royal Mail or Ordnance Survey.