I am trying to draw a timeline and I want to label an interval of time
using a horizontal double arrow such as <--->.
Unfortunately, because the nodes $t$ and $t + \ell_t$ have different heights,
the interval "Shipment lead time" is not exactly a horizontal line.
How do I fix the nodes so that they have the same height?
Instead of changing the nodes, you can use the perpendicular coordinate system (compare the red inclined rule with the two horizontal black rules):
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\node[anchor=south] (A) {\Huge A};
\node[anchor=south] at (2,0) (B) {B};
\draw[red] (A) -- (B);
\draw (A) -- (B.west|-A);
\draw (A.east|-B.west) -- (B.west);
\end{tikzpicture}
\end{document}
(<name1>|-<name2>) has the x-coordinate of <name1> and the y-coordinate of <name2>; (<name1>-|<name2>) has the y-coordinate of <name1> and the x-coordinate of <name2>.
If, however, you want the nodes to have the same total height, assign equal values to the keys text height and/or text depth:
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\node[draw] (A) {$t$};
\node[draw] at (2,0) (B) {$t_{l_{m}}$};
\draw[red] (A.south east) -- (B.south west);
\begin{scope}[xshift=3cm]
\node[draw,text depth=0.35ex] (A) {$t$};
\node[draw,text depth=0.35ex] at (2,0) (B) {$t_{l_{m}}$};
\draw (A.south east) -- (B.south west);
\end{scope}
\end{tikzpicture}
\end{document}
