I'm looking for some software - the requirements:

Preferably freeware, although having no idea what's on the market I'd
accept a freeware solution not being available.

Something that would be run on a machine with only some initial user
input that would trawl through a site at regular intervals and backup
everything locally. (Kind of like an offline mirror - but which would
only be made available to the web if the primary server went down).

The aim is to be able to simply switch the IP of the domain across to
the backup and continue as normal.

I realise some companies incorporate this kind of redundancy as part
of some hosting packages, but was wondering whether a solution was
already available in order to do this alone.

If You want something FREE, unix|Linux is a best choice.
An example of linux backup software is rsync.
But it is complicated OS in comparison with MS Windows.

http://www.linuxforum.com/shell/rsync/92-8.php
http://www.webmasterworld.com/forum40/907.htm

regards, s3com

I realise this might not be technically possible, but I was thinking
of a solution that would be more along the lines of inputting a top
level domain, and having something trawl the page for links/images
etc.. and saving this in the appropriate directory. This obviously
wouldn't work for php and such and would extremely limit the content
that would be backed up, but it wouldn't require the user/pass of the
server currently hosting the site which is what I'm initially after.

If you're familiar with linux, I believe the recursive "wget" command
can do such a trawl.  But I'm no expert.

Wget sounds like a perfect canidate for your requirements.  It has the
ability to do recursive crawls of linked files (which can be
restricted to all files on a given top-level domain, or other criteria
you specify).

Wget also has the ability to do http authentication, and ftp, and
local hyperlink conversion.  It has support for cookies; it has the
ability to

The main site for it is:
http://www.gnu.org/software/wget/wget.html
You can download it for windows, or several other OS's 

To do what you are saying you would need to add a scheduled task to
check the site on an interval.
The -N switch causes it to only download files if the site contains a newer one. 

A note on installation under windows:
There is no setup program, or msi package, but all you need to do is
download the wget-complete-stable.zip from the above site, and read
the readme.txt contained within.  I have been using it on several
windows systems for years now without any problems.  See the wget.hlp
from the wget-complete-stable.zip file if you need any more details on
this program.