Have a good look at the cover before taking it apart to work out how it fits together - make a sketch of it. It should be, as Eidsvold says, an oval for the base of the basket, a strip that goes round the inside sides and a looser, gathered bit that drapes over the outside sides. There may be some elsatic in there somewhere to hold it snugly on the basket.
Tell you what, if you're in the UK and you have a complete disaster, I'll sort it out for you! (see how confident I am that you'll manage it?! )
even cheaper and perhaps easier than JohnLewis ( only cause none close to us) - somewhere like tesco/woolworths I think I got my quikunpicker from
I am waiting to get some money together to buy a second hand sewing machine ( good one is back in Aus - not prepared to bring it to UK to then take it back to Aus) I have some brilliant patterns for clothes to sew for dd and myself!! Can't wait.
I had a similar issue and i did as the ladies above suggested even though I'm not the most cometant sewer in the world and it looks OK and wasn't too hard - it took me a weekend to do the baskety bit and I have to admit that i have not got round to doing the hoody thing.
I did get a set of instructions from a web search from one of the home type TV programmes - I can't find it at thh moment but it did make it more complicated than using a big piece of paper and copying the shape of the material i took off. I also changed any complicated bits like fold over flaps to openings and it looks great.
Firstly, do you remember what all the pieces are and where they went?? Try to make a note/sketch so you don't forget. Then you pin the pieces onto your new fabric and cut them out and "simply" sew them together.
I imagine there will probably be some gathering as the base of the basket is narrower than the top. When you pin the side piece(s) round the base, pin it at the top, bottom and them midway between (ie 12 o clock, 3, 6 and 9). The pin it midway between these points etc etc. It helps to get the wider fabric gathered evenly round the base.