Applying to PhD programs as a Masters student with no Internship experience
In the UK an internship is not necessary.  In general, an internship is just one way to prove yourself, a paper is another, but good letters of reference can compensate for these.  Generally it's good to have a good mix of skills in your lab.  Don't be defensive about it; just talk about your assets.  

On the other hand, if you don't get the PhD offer you want, you might find a research position (or a job) and get some additional research experience that way.  There's always next year for applications.