unicode-math but ordinary blackboard bold

\documentclass{article}
\usepackage{amssymb}
\let\mathbbalt\mathbb
\usepackage{unicode-math}
%\let\mathbb\mathbbalt% UNIVERSAL RESET TO ORIGINAL \mathbb
\begin{document}
\begin{equation}
\mathbb{R}\quad\mathbbalt{R}% OR JUST CALL ON INDIVIDUAL ORIGINAL GLYPHS
\end{equation}
\end{document}

As written above, one can access the original \mathbb glyphs via \mathbbalt.

Uncommenting the "UNIVERSAL RESET" line gives this result:

You can undefine \mathbb and then redefine it to do the same it does with amsfonts:

\documentclass{article}
\usepackage{unicode-math}

\AtBeginDocument{%
  \let\mathbb\relax
  \DeclareMathAlphabet{\mathbb}{U}{msb}{m}{n}%
}

\begin{document}

\[
\mathbb{R}
\]

\end{document}

You can still get the standard unicode-math bb letters by \symbb.

In addition to the previous answers (which did not work for me, since my unicode-math is outdated), I found the following workaround (I cannot install a newer version of unicode-math so easily, since I don't have administrator rights). What I did was insert the line

\setmathfont[range=\mathbb]{texgyrepagella-math.otf}

This gave me (despite a warning that the font is not found and "defaults are substituted"), the nice \mathbb{R} I wanted (no idea what the substitute font is called, but it's obviously the right one).

Thanks again also for the other answers which are, of course, the way to go on an up-to-date installation.