According to the latest Ofcom report, here, the most popular news brand on mobile handsets was the BBC (14.5 million visitors).
Sky had 4.2 million visitors,
Mail Online 3.4 million visitors,
The Guardian 2.0 million visitors.

The BBC was up 1.3 million, year-on-year.

The Ofcom report is wide-ranging, and covers much internet-connected data.

Section 4.37 "On-Line News"
Figures 4.59 and 4.60 refer specifically to mobile news consumption.

The report also notes the drop in laptop / desktop use, and the BBC's position:-
The Daily Mail - 10.6 (down 0.4)
The Guardian - 10.6 (down 0.2)
The BBC - 10.3, (having lost 11% (1.3 million visitors) year-on-year).



(Thanks to Journalism today.)