"I may be able to stretch this to an absolute maximum of £400."
That's a popular price point,and there is a wide choice, but the ideal situation is to find a machine that was previously sold at a higher price. With that in mind, take a look at this
Lenovo machines are generally well-built - I have one myself - and this one ticks the boxes. It has 4 USB port, and will easily cope with the use you plan to put it to.
You'll connect your external drive via one of the USB ports.