I am trying to convert the CSV file that I have for a shapefile. The CSV file contains a Cartesian xy coordinate field that I wanted to convert to xy longitude and latitude field in the shapefile. I have QGIS I succeeded in converting the CSV file to shapefile but the coordinate field is the same xy Cartesian. I am attaching the files I want to convert in the comment, two of them are point data and one is polygon data. All the files contain data for places in the UK.
You need to bring in the data as you say in "cartesian" coordinates. Most likely your data is in British National Grid, so EPSG:27700. But you want to check this in the source data.
Once it is in QGIS, you need to reproject the data (convert to a different projection).
This can be done with Vector> Data Management Tools> Reproject layer.
And reproject to WGS:84, EPSG:4326. Which will transform the coordinates to Lat/Long.