I've been running apt-get every couple of days since I started using debian but for the past couple of weeks I've been getting this error -
Failed to fetch http://ftp.uk.debian.org/debian/dist...86/Packages.gz Connection failed

This has happened before but has never gone more than a couple of days before going back to normal.
I tried changing from ftp.uk.debian to ftp.de.debain which has always worked before when there have been problems but that is also giving the same error.
There are no other errors from the other repositories but there hasn't been a single update available for any program at all during these last two weeks.
Normally there are quite a few after only a couple of days.

So is this a problem on my computer or have repostitories been changed and how would I find out what new ones to use?
Or is there a problem with debian servers at the moment or maybe there just hasn't been any updates?

No, there are no problems at the moment, at least not as far as I can tell. I've been getting the normal number of updates the last days and that particular URL doesn't give me any trouble at all, although it's not what I have in my sources.list. (I'm using the repository at http://ftp.sunet.se/pub/os/Linux/distributions/debian/.) So it really looks like you have some kind of error at your end. I'm sorry I can't tell what it might be...

Well I tried another apt-get update so I could check all the wording of the error messages and as if by magic, it ran all the way through without a single error.
I now have a nice big list of updates for my dial-up modem to groan at.

It is not uncommon for some of the Debian mirror sites to have problems from time to time. If one doesn't start working again in a day, try browsing into it with a browser because sometimes all the problem is that they moved things around a little bit and you just have to modify your sources.list. Otherwise, try another Debian site close to you.