You can override various time-returning functions via the mechanism documented in perlsub:

package SetTime;
use warnings;
use strict;
use Exporter ();
our @ISA = 'Exporter';
our @EXPORT = qw/time localtime gmtime set_time/;
our $VERSION = 0.01;

sub import {
    my $pkg = shift;
    $pkg->export("CORE::GLOBAL", "time", "localtime", "gmtime");
    $pkg->export(scalar(caller(0)), "set_time");
}

my $time;
sub set_time { $time = shift }

sub time () {
    defined($time) ? $time : CORE::time;
}

sub localtime (;$) {
    @_             ? CORE::localtime($_[0]) :
    defined($time) ? CORE::localtime($time) :
                     CORE::localtime();
}

sub gmtime (;$) {
    @_             ? CORE::gmtime($_[0]) :
    defined($time) ? CORE::gmtime($time) :
                     CORE::gmtime();
}

1;
[download]

But beware of conflict with other modules that override these (offhand, Time::localtime and Time::gmtime do this), or other ways of getting the time (POSIX, or other XS modules).