The receptionist probably gets loads of enquiries and will just give you a stock answer to get rid of you 
The best thing is to write to one of the partners. Do a really good letter that you can reuse to send to other practices, and add in your CV. See it as a job application.
Introduce yourself, make your request, outline your reasons for wanting WE and convince them of why you would be a help rather than a hindrance.
Be realistic about your expectations of what you will be doing (observing, making tea and cleaning up poo) and make it clear how grateful you would be and how much of a help it would be to you when applying for uni.
Make a note of when and where you send each letter, then after a week follow it up with a 'hello, did you get my letter?' phone call. Ask to speak to the person you addressed the letter to (a partner). Chase (politely) until you can speak to them. Make it clear that even a day, even a few hours, even 1 hour would be amazing - once they meet you and you wow them you will have more doors opened to you.
If you have no joy, ask the partner when you speak to them if they can recommend any farms on their books who take work experience pupils. Vets are usually only too happy to help out aspiring vets, but you must be keen. One phone call to a receptionist just doesn't cut it, as you've found.
Ask your parents to scour their memories and address books for any contacts, any at all who work at vet practices. A lot of the time, whether it's right or not, who you know makes a difference.
It is possible, and if you want it enough you'll get there.
