Essential Aearodrome Information should be put on the ATIS to prevent ATC having to read it out each time to each aircraft (go look at the MATS Part 1 on the CAA website if you want to check)
The Muir Matheson system in use at Bristol is also in use at most regional airports in the UK and would have similar functionality, whether the local ATC use these features is of course up to them (and the extent of weather conditions affecting the particular airport concerned) but you can be assured that it would be SRG approved and certified with the relevant safety cases considered and inclusion of Met warnings on the ATIS would be on the specific direction of the Met Office.
Furthermore, Bristol Airport does not have a Met Office, the nearest being in Cardiff (in the city not the airport) who use the METARs provided by observers at Bristol and other information to formulate forecasts and warnings. Therefore Bristol cannot be responsible for the specifics of the warnings, they are simply passing on potentially useful information as they are required to do. If you want clarification of any weather conditions present or forecast then the weather centre is only a phonecall away.
The ATIS message is still structured and concise even with warnings appended, if you're not interested then don't listen but please do lay off ATC for doing their job as directed.