Twitter will not charge you for text messages, or to use mobile.twitter.com. However, you may see usage charges from your service provider depending on the type of data/rate plan you are currently on.
Your carrier will charge you for text messages from Twitter in the same way as text messages from someone you know. For example:
Note that standard rates do apply when you add your phone to Twitter. Your service provider might have fees for sending and receiving SMS/text messages or using your mobile browser to connect to Twitter.
Check here to see if your specific carrier allows two-way (sending and receiving) service with Twitter. If you don't see your country-specific carrier on that list, you can still use Twitter with your phone! Once your phone is verified, you'll be able to send your updates to +44 7624 801423, but you will not be able to receive text updates from Twitter.
Some carriers, even in the UK, consider Twitter's long code to be an international number and charge from .15 to .25 per message. Check with your carrier for details.