To align the baseline of the text in the inner node on the baseline of the outer text you can use \tikz[baseline,nodes={anchor=base,...}] for the inner picture.
To prevent the influence of the outer node options save the inner picture in a box:
\documentclass[12pt,tikz,border=1mm]{standalone}\usetikzlibrary{chains,shapes.multipart}\newbox\mytikzBox\newcommand{\savetikzBox}[2][\mytikzBox]{% \sbox#1{\pgfinterruptpicture#2\endpgfinterruptpicture}}\begin{document}\begin{tikzpicture} \savetikzBox{% \tikz[ baseline, start chain = going right, nodes={anchor=base, fill=gray!10, inner xsep=0.5mm,outer sep=0mm, on chain} ,node distance=.5mm ] {\node{0};\node{0}; \node[fill=none,inner xsep=0pt]{:}; \node{0};\node{0};} } \node[ rectangle split,rectangle split parts=2, inner ysep=2mm, text width=55mm, draw] {adjust time\qquad\usebox\mytikzBox \nodepart{two} only text };\end{tikzpicture}\end{document}