I have got a berchet pram from e-bay. They are no longer made but you can always get them on e-bay here I hope
This one has a frilly cover on but you can just take it off. They are really attractive to look at and low enough for a one year old.
Or there are those 'Lil lady ladybird' prams on e-bay too.
Before I got my lovely Berchet pram I had a plastic one from mothercare and it was rubbish. I took it back.
They do love cheap buggies the most but I would get a pram for a one year old as they are more stable. My dd plays with hers all of the time, she got it when she was one.
My neiece has got hat wooden one from your OP and it it too heavy.