Welcome to Taiwan!!!
It's quite easy to get to Taiwan if you are OK to take a long journey.
You can either take one of two Taiwanese airlines- Eva Airways & China Airline, or other airlines such as Cathay Pacific, and British Airways. Of course your may also check your own national airlines. The easiest way to purchase flight tickets from the UK is to visit the Taiwanese Travel Agency-Our Travel; or check opodo for bargain prices.
To get brief travel information, you can visit here for suggested schedules or advice.
And if you have any further concerns, please take a look at the Taiwanese Tourism Bureau.
Or...you can write to me!