Beats Flex are not charging

Let's start with the charging issue. For that the following support articles should prove helpful:

• Beats Repair & Service - Apple Support
• Beats support: International numbers - Apple Support

Is it possible to get the battery replaced if the problem persists?

I got my headphones as a freebie from my phone company so I am not sure if they are under warranty. I am based in the UK.

You certainly can have them looked at to see if a battery replacement is needed. However, since you got them as a "freebie," getting the replacement will most likely have a cost involved. Apple typically warrants their devices for one year against manufacturing defects. If you got these within one year of when it was actually purchased by your phone company, they may have the grounds to get warranty work done ... but this would be an unusual case, and I suspect, you will have to "foot the bill" for any repairs.