For the internet part, as Nick wrote, I'd check ThinkBroadband.com as you can compare different ISPs and see graphs for the last 6 months showing reliability, customer service, and speed. What speed you get will depend on how far from the exchange you are (unless you have the option to use Virgin cable) and the range of deals you can use will depend a bit on where you live... In most cities you have more choice than somewhere rural, and the costs are often lower, with greater usage allowance.
You are clearly already using the internet, so do you have to give some proof of charges to get a portion of the grant for internet access? Since there are many options when it comes to internet access, do you already know what amount of traffic your family uses (ie if the ISP offers 'starter' 'medium' and 'premium' accounts giving say 2 GB, 10 GB and 30 GB of data, do you know which you use?)
I only ask because with a new machine there may well be a lot more downloading - assuming you won't just 'scrap' the current PC - as you'll have 2 PCs to use in future...
As for a new PC, in the past I've suggested any client visit places like PC World and Comet to try them out, but now (and especially with an expensive new machine) I'd say it is probably just as easy to have separate USB keyboard and mouse (or even, since you have a generous budget) wireless mouse and keyboard - they don't cost the earth - USB multimedia keyboard at a tenner or less, and optical (laser) mouse at under 4 quid inc delivery (eg from Ebay or Amazon).
Why bother with separate keyboard and mouse? Well, all too often someone manages to spill coffee or wine or lemonade into the keyboard section of a laptop and can pretty easily destroy it when powered up... There's no need to have the machine right up to your face when it has a 15+" screen and you can 'share' the view more easily if it is just that bit further back on a table/ desk. Mouse is probably a lot better than the built-in pad / trackball / joystick and a full size keyboard is cheap to replace if it ever gets damaged... a multimedia keyboard has the 'play' 'next track' 'mute' 'vol+' 'vol-' buttons as well as the regular keys, and some have a 'sleep' button too, so you can make the laptop 'sleep' and turning it back on, it will have exactly the same windows open, and everything as it was when told to sleep.
I know PCs will run faster with more RAM, but for general use, rather than by a 'power' user, 2 GB should be OK (I recently bought a Compaq laptop at 250 quid with Windows 7 on, and it is very responsive). If you do want to go for something with plenty of RAM, medium (in today's terms) hard drive, and don't mind spending nearly 400 quid, then THIS might do all you need.
It's the big sister Compaq (I bought mine in Asda Living, where the price had been dropped by about 100 quid), but at 379, this should tick most boxes, as it has 4 GB of RAM, 250 GB Hard Drive, and widescreen (but can be plugged into a big plasma or LCD TV using HDMI if it was used for showing holiday photos, or perhaps running some games).
As I'm giving my sister the one I bought (Asda sold out) I am even tempted to get this one for myself in place of it!!