All employees have a contract with their employer even if nothing is actually written down. What you're referring to is a 'written statement of employment particulars'. Your original employer was obliged to provide you with such a statement within two months of starting employment. (That document doesn't have to be called a 'contract'. It could simply be the letter offering you the job as long as all relevant matters were covered within that letter).
Your new employer should also have provided you with a written statement of employment particulars but, once again, there was no need for it to be worded as a formal 'contract'. A letter stating that all of your contractual rights and duties, inherited from your previous employer, were still in place would be sufficient to meet the legal requirements.
Chris