After looking around and not finding any decent script to read the weather for Asterisk, I decided to write my own.
The code is downloadable here. Save the file to /var/lib/asterisk/agi-bin/getweather.agi.
To use this script in Asterisk, copy the following into /etc/asterisk/extensions.conf:
exten => 997,1,Answer
exten => 997,2,agi,getweather.agi
exten => 997,3,Hangup
Update 6/6: I've updated this script to v1.1 now. I've fixed up a parsing error and also added more wind directions - the BOM add more than I thought ;)