relating two diagrams in tikzcd

To create the arrow between arrows, just add empty labels, they create nodes you can later refer to draw the arrow, see package documentation at Section 2.3 Absolute placement of arrows.

Moreover, you don't need to add empty rows and columns to increase the row/column sep, there are appropriate options: row sep=..., column sep=....

Edit: to add an implication arrow as in $\Rightarrow$, just add Rightarrow option, see package documentation at Section 1.3 Changing arrow tips.

\documentclass[12pt,reqno,a4paper]{amsart}
%\usepackage[utf8]{inputenc} % not needed with up-to-date TeX systems
\usepackage[english]{babel}
\usepackage{extsizes}
\usepackage{blindtext}
\usepackage{amsmath,amsthm,amsfonts,amssymb}
\usepackage{mathtools}
\usepackage{color}

\usepackage{mathrsfs}
\usepackage[all]{xy}
\usepackage{tikz-cd}
\usepackage[normalem]{ulem} % do you *really* need it?
\usepackage{datetime}

\usepackage{hyperref}

\begin{document}

\begin{tikzcd}[row sep=2cm, column sep=2cm]
A \arrow[d] \arrow[r] & B \arrow[d,""{name=BF}] &  C \arrow[d,""{name=CD, left}] \arrow[r] &  D \arrow[d] \\
E \arrow[r]  &  F &  G \arrow[r]  &  H 
\arrow[from=BF,to=CD, Rightarrow]
\end{tikzcd}

\end{document}

You only need to ad invisible characters to cell in the middle row before arrow and in cell after arrow:

\documentclass[12pt,reqno,a4paper]{amsart}
\usepackage[english]{babel}
\usepackage{extsizes}
\usepackage{blindtext}
\usepackage{amssymb,amsthm}
\usepackage{mathtools}
\usepackage{color}

\usepackage{mathrsfs}
%\usepackage[all]{xy}
\usepackage{tikz-cd}
%\usepackage[normalem]{ulem} % do you *really* need it?
%\usepackage{datetime}

%\usepackage{hyperref}

\begin{document}
    \begin{tikzcd}[row sep=1cm, column sep=2cm]
A \ar[dd]\ar[r] & B \ar[dd] & C \ar[dd] \ar[r]  & D \ar[dd] \\
    & ~\ar[r,Rightarrow]    & ~                 &           \\
E \ar[r]        & F         & G \ar[r]          &  H    
    \end{tikzcd}
\end{document}