"we were hoping to build a timber framed extension, something with barn like elements, and I wonder if anyone has undertaken this, and whether it's feasible to get the timber frame company to design it rather than using an architect?"
So you're looking for a traditional oak framed extension, rather than a timber frame in the modern sense?
Yes, I've done them, but yes, it's certainly also possible (and far from a bad idea!) for the specialist timber frame company to undertake the design work for you.
If you chose to go down this route, I'd recommend talking to Oakwrights as a very trustworthy and competent contactor (no connection other than I've worked with them on past projects and have been impressed by their competence):
www.oakwrights.co.uk/
This is a specialist construction technique, with a lot of specialist knowledge attached to it, so make sure that whoever does the design work is genuinely experienced in it. And expect the fees to be quite a bit more than stated in your original post - sorry, I was assuming you were talking about modern timber frame construction.
For reasons I won't bore you with, it's getting very difficult to gain Building Regulations approval for new build properties in oak frame where you can see the fame both internally and externally, in the old traditional manner, but it's still just about possible with an extension, where lower standards apply.
Make sure you (and more importantly your Architect) understand issues like shrinkage if you're working in green oak, though you can minimise these using Glulam instead of oak... otherwise, you've got to recognise that the limitations are part of the character of the material.