I don't like using chemicals, so if I had to deal with that I'd invest in some tough gardening gloves, cover my arms and dig them up, so the roots were removed.

Hi Lynne and welcome to GoY digging out really is the best thing to do, roundup wont really deal with thistles.

It's only a question for you of how much you'd like the allotment - there will always be some nasty weeds to deal with - I'd say go for it!
Why not get some friends to help so the job isn't so daunting. Post a blog on here with photos to show how you get on - ? You'll find that digging them up after rain is much easier than when the soil is dry, if you get the chance that is. Good luck, hope it goes well for you. : o )

I would take the allotment if I was you and do what weeding suggested have bee on a waiting list for several allotments in the area I live for about five years and I will be waiting for at least another five years to get one

Oh lord missed that - yes definitely take the allotment!

Try an allotment clearing party. BBQ etc. Provide gloves and promises of fresh veg might be an incentive to some people. Bring own tools and mind your backs.