When I watched the series I was so struck by this segment that I went looking for the soundtrack. It's called "New Queen," with apparently semi-ironic intent, since it occurs at the end of Series 3, Elizabeth's Silver Jubilee.
I absolutely love that piece. I just wish it went on longer. There are other good things in the soundtrack but nothing grabbed me as much as this.
Here's the whole scene. I take it for granted that the series gives a picture in some ways false, but whatever might be said along that line, Olivia Colman's performance as Elizabeth is outstanding: