OK, we can already see the first solution: slow down. :)  As a good speaker, native or not, you should be able to control the pace of your speech.  I'm sure your accent isn't clear because you're speaking too fast.

If you want a proper accent, then you need to interact, not only imitate.  Find people who speak the way you want to sound, and work with them. Consider getting a teacher for a couple of lessons, just to guide you.  If you search through the italki teachers, you will find several teachers from the UK who can specifically guide you towards your goal.  Even just one session may open up a load of possibilities.

For getting a British accent you have to listen to British people and try to imitate them. You can watch videos on YouTube. I know some very good channels to practise the listening.

https://www.youtube.com/channel/UC0Hg2Ks00kCekyjZG_LxOmg/videos

https://www.youtube.com/channel/UCz4tgANd4yy8Oe0iXCdSWfA/videos

https://www.youtube.com/user/papateachme/videos