Eulervm and LuaLaTeX results in wrong front for math operators

You can use legacy fonts for the operators:

\documentclass[]{article}

\usepackage{amsmath}   
\usepackage{ebgaramond}
\usepackage[euler-digits,small,OT1]{eulervm}

\usepackage{fontspec}
\DeclareSymbolFont{operators}{OT1}{EBGaramond-TLF}{m}{n}

\begin{document}
\begin{equation*}
1+ \sin \hat{x} \tilde{x} \quad \text{sin}
\end{equation*}

\end{document}

Alternatively, use the legacy font only for the accents.

\documentclass[]{article}

\usepackage{ebgaramond}
\usepackage[euler-digits,small,OT1]{eulervm}
\usepackage{fontspec}

\DeclareSymbolFont{operators}{\encodingdefault}{\familydefault}{m}{n}
\DeclareSymbolFont{accents}{OT1}{EBGaramond-TLF}{m}{n}

\DeclareMathAccent{\acute}{\mathalpha}{accents}{"13}
\DeclareMathAccent{\grave}{\mathalpha}{accents}{"12}
\DeclareMathAccent{\ddot}{\mathalpha}{accents}{"7F}
\DeclareMathAccent{\tilde}{\mathalpha}{accents}{"7E}
\DeclareMathAccent{\bar}{\mathalpha}{accents}{"16}
\DeclareMathAccent{\breve}{\mathalpha}{accents}{"15}
\DeclareMathAccent{\check}{\mathalpha}{accents}{"14}
\DeclareMathAccent{\hat}{\mathalpha}{accents}{"5E}
\DeclareMathAccent{\dot}{\mathalpha}{accents}{"5F}
\DeclareMathAccent{\mathring}{\mathalpha}{accents}{"17}

\usepackage{amsmath}

\begin{document}
\begin{equation*}
1+ \sin \hat{x} \tilde{x} \quad \text{sin}
\end{equation*}

\end{document}