I would see if you can cancel the contract with them, get a refund, after finding out who does definitely have it in stock (John Lewis, Mothercare, Kiddicare are all good websites - not sure what travel system you have ordered but one of them should be a supplier)
If you have trouble cancelling the contract and getting a refund, call consumer direct for advice - they have to refund you within 30 days.
Ideally you'd have a credit card or enough money to buy it "again" with a more reputable website whilst chasing your refund - you are covered under the distance selling regulations and consumer credit act, so this definitely will happen - then at least you have your product, and you can chase the refund seperately.
Always buy from John Lewis or Mothercare if they have it in stock - both are excellent at observing the Sale of Goods act, have 28 day returns (or more sometimes I think), John Lewis has a price promise, "never knowingly undersold", and both if you buy from the website have records under the "your account" going back years so you can go back in 3 years time if it breaks and print out a proof of purchase, take it back to the shop, and get it repaired or replaced, because both shops would rather just do that than fight you for it and get a bad rep.
Good luck