You could use freely available Code-Point® Open dataset from Ordnance urvey (available from here):
Code-Point Open provides a precise geographic location for each
postcode unit in Great Britain. The product is a CSV file containing
postcodes, grid references, NHS® health and regional health authority
codes, administrative ward, district, county and country area codes.
It will provide you with csv file of postcodes' (centroid I think) Eastings and Northings from which you can create XY feature class.
Note: These are actually separate files for each one of the postcode areas in UK. You can quickly merge them into one file from command line using:
copy PATH_TO_YOUR_DATA\*.csv PATH_TO_YOUR_DATA\OUTPUT_FILE.csv
Then you could merge your postcodes to that dataset to keep only relevant postcodes.
Bear in mind that postcodes in UK are 'living things' - they do change in time, so make sure to check if that is the case for any of the postcodes from you data. And make sure to review Ordnance Survey's licence for the data.