This sounds like a problem with the current server then, so I've placed an order for a much more powerful one which should be online in the next day or so. It will then take a couple of days of testing, so I guess I will be able to move the site over in the middle of the week.
I don't expect there will be more than a few minutes of downtime (hopefully none at all), but I will post here as the transfer is about to happen.
The new server has 12GB Ram and dual Xeon 5520 CPU's, so it should fly

I was going to go for the Intel SLC SSDs for the database drive, however I'll see how the 15k SAS drives work for now. It will be easy enough to upgrade the drive if required.