Calligraphic or handwriting fonts that use diacritics

Of the type 1 fonts listed in the font catalogue, miama appears to work:

\documentclass{article}    
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{miama}    
\begin{document}
\fmmfamily
Quō ūsque tandem, Catilina, abūtēre patientiā nostrā?

Qu\=o \=usque tandem, Catilina, ab\=ut\=ere patienti\=a nostr\=a?
\end{document}

The TeX Gyre version of Zapf Chancery also works:

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{tgchorus}
\begin{document}
Quō ūsque tandem, Catilina, abūtēre patientiā nostrā?

Qu\=o \=usque tandem, Catilina, ab\=ut\=ere patienti\=a nostr\=a?
\end{document}

If you're free to use either XeLaTeX or LuaLaTeX, there are literally hundreds, if not thousands, of chancery-type and handwriting-type fonts out there that are set up to handle diacritics.

The following example shows just two such possibilities: Apple Chancery and Zapfino. (Times New Roman is shown mostly for contrast.)

\documentclass{article}
\usepackage{fontspec}
\setmainfont{Times New Roman} % set a suitable default text font
\defaultfontfeatures{Scale=MatchUppercase}
\newfontfamily{\AC}{Apple Chancery}
\newfontfamily{\zapfino}{Zapfino}
\begin{document}
\obeylines % just for this document
Times New Roman
Quō ūsque tandem, Catilina, abūtēre patientiā nostrā?
\bigskip Apple Chancery
{\AC Quō ūsque tandem, Catilina, abūtēre patientiā nostrā?}
\bigskip ...
\bigskip Zapfino
{\zapfino Quō ūsque tandem, Catilina, abūtēre patientiā nostrā?}
\end{document}