How important is it to choose and stick to one writing style (Oxford, Associated Press etc.)?

The writing style depends on your subject.
Oxford (academic style) is reserved for essays/books/magazines and basicly academic dissertations.
While Associated Press (news agency style) is used for short articles/news and is focused clearly on the message you want to get with to your audience.
When you write a news it is essential to use short sentences and squeeze as much content as you can in small amount of words...You also have to answer in first two sentences- who,what,where and when happened a situation you writing about...Every other sentence should be additional what means that cutting words/sentences from the bottom of your text should not deprive it the meaning/your message.
The academic style can be more decorative and colorful- it is not only important what you writing about but also how you writing (what is your style).
In news agency style you rather don't use adjectives.
Choosing a right style and not mixing it with another evidence your consistency and easiness of reading.

well, you should try to stick to one writing style so that you will always have a certain way that is well known to you...
but sometimes when you are writing an essay you may be restricted to using a way that isn't known to you...
this happens when the side you are writing to tells you what to use in your writing...
:)