Please, I need to prepare a map for the study area of the United Kingdom, what is the correct projection to choose in ArcMap? Should I use WGS84 and 31 N ? Could anyone advise me please, many thanks.
There are a few you can use. I would say it depends on what the client wants.
You could try the National Grid of Great Britain - EPSG:27700 (not the best for Northern Ireland but grabs the 'big' island.
There are some ideas on this page: https://epsg.io/?q=United%20Kingdom%20kind%3APROJCRS
I totally agree with my fellow Aggie @AdrianWelsh (I'm class of 81 - Gig'em) BNG (EPSG:27700) is the Coordinate system to use for mapping GB. Here is some additional documentation should you require further detail: UK Geospatial Data Standards Register, and, from Wikipedia, Ordnance Survey National Grid. If you really are into Geodesy and Positioning you will appreciate A Guide to Coordinate Systems in Great Britain.
Hope the above helps.