It is worth wearing technical clothing for that length of journey, but it doesn't have to be race-style Lycra gear. There are lots of options for commuting wear. Even then, you don't have to stick to bike-specific kit.
You can actually get padded boxer shorts (
Evans do them IIRC) which fulfil
part of what cycling-specific shorts are about. You can then wear whatever trousers or shorts you like. That's only part of the story because there's the warmth/waterproof side of things too. In my experience, on a 15 mile ride you'll get fairly warm (depending on weather and your speed) so you might want some Lycra but if you really feel self-concious then anything that won't get saturated if it rains (like cotton or denim) or caught in your chain will do. For 15 miles (I'm guessing about an hour), I reckon if it rains you're better off with Lycra or something that will get wet but not saturate. Wearing waterproof trousers will get you just as wet from sweat on the inside.
On the top any tech tee will do, cycling jerseys are obviously made for it but any synthetic wicking tee will do, such as the cheapo Peter Storm ones from
Millets. These are cheap enough that you can buy 5 - a fresh one for each day. Then all you need is a wind or rain jacket if it's not so nice out.
If you haven't got a shower at work, just make sure you change out of your teeshirt and into the work kit as soon as. Sit there for an hour checking your email before changing and you might give rise to nasty niffs. Also recommended is these salt-type deos such as
Ice Guard which you can get from Holland and Barrett. They don't leave a mess and basically stop bacteria, so you can sweat as much as you like without getting BO. Plus a stick lasts absolutely forever for your fiver.
I remember a guy who used to cycle 12 miles to work and wear his work shirt on the ride and all day. That was 25 years ago and I still remember him, the stinky old hippy. You don't want to make that kind of impression on anyone!