For £100 look for a reconditioned or second hand IMHO. There are some outfits (like Edinburgh's Bike Station) who rescue unwanted bikes and sell them on for bargain prices: highly recommended with only £100 to spend. Check to see if someone does this in your area.
Mountain bikes are designed for off-road, which is Good, but are consequently less good on it, which is Not Good. You can make life a lot better with a set of slick tyres, but a decent pair will be £30+ so if I were you I'd avoid MTBs, or (at this price) MTB-a-likes.
You're right about mudguards, and also about suspension. You want to avoid discs for the same reason as suspension: done well they're great, but done well costs Real Money (TM) and I'm afraid we don't have that here.
If you're upping your budget (which I'd heartily recommend) I'd say don't go to Argos. Go to a bike shop, where you can try things out for size and feel and talk to someone who knows what they're about. Halfords may or may not count, it's very much down to the staff in individual stores.
For a ten mile commute a rack and panniers for any luggage is a lot better than a rucksack, so add budget for rack as well as mudguards. And don't forget lights too...
A no-frills hybrid should be workable from about £150 (before you spend your extra on ;guards and rack) IMHO, but if you pay twice as much as that you'll get a lot more for your money. The law of dimimishing returns will start to cut in above that. I think Guy's starter level of £200 is a good one. You can buy a bike for £100 but 10 miles a day will soon have inferior components suffering, and you along with them. And if you buy cheap and nasty to upgrade later, the frame is the key to the bike so it'll not really be worth it: you won't turn a Fiesta Popular in to a Ferrari by adding cool mirrors and alloy wheels...
Pete.