Your Japanese sentence is understandable.
And Atsu-san's sentence surely sounds natural.
But if you'd like to get some idea to connect sentences, my sentence would be this, using て.:
私はイギリスに住んでいて、独学で日本語を学んでいる/います。
or just simply
私はイギリスで独学で日本語を学んでいる/学んでいます。