@John: Visually, there may not be anything other than TNR. Meferdati's problem seems to be other fonts being embedded in the pdf.
Now the reason they are embedded becomes clear when you minimze your minimal example further:
\documentclass{scrartcl}
\usepackage{tikz}
\usepackage{pgfplots}
\usepackage{verbatim}
\usepackage{mathspec}
\setallmainfonts{Times New Roman}
\setallsansfonts{Arial}
\setallmonofonts{Courier New}
\begin{document}
% \pgfplotsset{every axis plot post/.append style={line width=1.0pt}}
% \pgfplotsset{grid style=dotted}
\begin{tikzpicture}
% \begin{loglogaxis}
% [clip marker paths=true,
% xlabel=background field amplitude $B_{\text{a}}$ / T,
% ylabel=losses $Q$ / J/cycle/m,
% xmin=2e-3, xmax=5e-2,
% ymin=1e-4, ymax=1e-1,
% xtick={2e-3,5e-3,1e-2,2e-2,5e-2},
% xticklabel style={/pgf/number format/.cd,fixed,precision=3},
% xticklabel={%
% \pgfmathfloatparsenumber{\tick}%
% \pgfmathfloatexp{\pgfmathresult}%
% \pgfmathprintnumber{\pgfmathresult}%
% },
% minor xtick={3e-3,4e-3,6e-3,7e-3,8e-3,9e-3,1.1e-2,1.2e-2,1.3e-2,1.4e-2,1.5e-2,1.6e-2,1.7e-2,1.8e-2,1.9e-2,3e-2,4e-2},
% grid=major
% ]
% \end{loglogaxis}
\end{tikzpicture}
\end{document}
tikz simply uses fonts from the Computer Modern series to produce the lines and dots in the diagram. IMHO the best solution is not to worry about that. Not only (as Ulrike said) is it highly unlikely that a Physics journal is not going accept a document that uses the math font par excellence. The point is: even if they wanted consistent font choice across all articles and images for a uniform look (which is a good idea), your image won't be a problem because Computer Modern isn't used for anything but lines and dots. Plus, the fonts are embedded in the pdf, so they don't need to have the font files on their machines in order to print it (this is a problem sometimes in print production).
I'd just submit it as it is and wait for their response. If it's rejected, you can either look for a way to make tikz use something else for dots and lines, or (the easier way) convert your diagram (or parts of it) to plain paths so no fonts get used at all.
Best Answer
Starting with LyX 2.2.0, the file in-between a .lyx file and the output .pdf file are more accessible. To get all of the files, do the following:
A directory will pop up with all files that were used to create the PDF.
If you happen to be using Linux, these files can alternatively be located by just navigating to the folder /tmp/lyx_tmpdir*/lyx_tmpbuf0 or something like that. I'm not sure how it works on Mac or Windows.