The following redefinition in your preamble ought to be enough
\DefineBibliographyExtras{american}{%
\protected\def\mkbibdateapalongmdy#1#2#3{%
\iffieldundef{#3}
{}
{\mkbibordinal{\thefield{#3}}%
\iffieldundef{#2}{}{\nobreakspace}}%
\iffieldundef{#2}
{}
{\mkbibmonth{\thefield{#2}}%
\iffieldundef{#1}{}{\space}}%
\iffieldbibstring{#1}{\bibstring{\thefield{#1}}}{\stripzeros{\thefield{#1}}}}}
but since the current version of biblatex-apa used \DefineBibliographyExtras instead of \DeclareBibliographyExtras in its .lbx files that does not quite work. This bug has already been resolved in the github repository, it has not made it to CTAN yet, though.
As a temporary work-around I suggest you create a new file called american-bpb.lbx and fill it with
\ProvidesFile{american-bpb.lbx}[2015/10/10\space v6.7\space APA biblatex localisation - modified]
\InheritBibliographyExtras{american-apa}
\DeclareBibliographyStrings{inherit = {american-apa}}
\DefineBibliographyExtras{american}{%
\protected\def\mkbibdateapalongmdy#1#2#3{%
\iffieldundef{#3}
{}
{\mkbibordinal{\thefield{#3}}%
\iffieldundef{#2}{}{\nobreakspace}}%
\iffieldundef{#2}
{}
{\mkbibmonth{\thefield{#2}}%
\iffieldundef{#1}{}{\space}}%
\iffieldbibstring{#1}{\bibstring{\thefield{#1}}}{\stripzeros{\thefield{#1}}}}}
\endinput
In your document then use \DeclareLanguageMapping{american}{american-bpb} (instead of the normally required \DeclareLanguageMapping{american}{american-apa})
In the MWE we create the american-bpb.lbx via filecontents
\documentclass{article}
\usepackage{filecontents}
\usepackage[american]{babel}
\begin{filecontents}{american-bpb.lbx}
\ProvidesFile{american-bpb.lbx}[2015/10/10\space v6.7\space APA biblatex localisation - modified]
\InheritBibliographyExtras{american-apa}
\DeclareBibliographyStrings{inherit = {american-apa}}
\DefineBibliographyExtras{american}{%
\protected\def\mkbibdateapalongmdy#1#2#3{%
\iffieldundef{#3}
{}
{\mkbibordinal{\thefield{#3}}%
\iffieldundef{#2}{}{\nobreakspace}}%
\iffieldundef{#2}
{}
{\mkbibmonth{\thefield{#2}}%
\iffieldundef{#1}{}{\space}}%
\iffieldbibstring{#1}{\bibstring{\thefield{#1}}}{\stripzeros{\thefield{#1}}}}}
\endinput
\end{filecontents}
\usepackage[style=apa, backend=biber, dateabbrev=false]{biblatex}
\DeclareLanguageMapping{american}{american-bpb}
\begin{filecontents}{\jobname.bib}
@online{online,
title = {Something on the Internet},
author = {Anne Uthor},
date = {2015-04-08},
url = {http://www.example.com},
urldate = {2015-08-04},
}
\end{filecontents}
\addbibresource{\jobname.bib}
\begin{document}
\nocite{*}
\printbibliography
\end{document}

biblatex-apato function one always needs the line\DeclareLanguageMapping{<language>}{<language>-apa}, but this is not available foraustralian(australian-apa.lbxdoes not exist). I think it might be easier to solve your problem in a different way. What exactly do you want to achieve? – moewe Apr 10 '15 at 05:23americanonly (and the necessary\DeclareLanguageMapping{american}{american-apa}) I get "(2005, February 14)" in the output. – moewe Apr 10 '15 at 09:01