There are two slightly different types of question in play here: questions of principle and questions of evidence.
The questions of principle are: (A) if matters transpired as described, was the first contract properly formed; and (B) did the second contract incorporate the schedules? With respect to the first contract, the absence of the quoted wording would not in itself invalidate it, and it seems likely that the contract was effectively executed, although I couldn't be 100% sure without knowing all the facts. In English law, most types of contract do not need to to be in a particular form, or include particular forms of words, and indeed in principle it is possible to form a contract with a handshake, or any other communication of acceptance. With respect to the second contract, the emails attaching signed schedules bouncing back would not in the usual course of events mean that the schedules are not part of the contract.
The questions of evidence are: (A) if you needed to prove in court that the first contract was properly formed, would you be able to do so on the balance of probabilities; and (B) if you needed to prove in court that the schedules formed part of the second contract, would you be able to do so to that standard? In both cases the most likely answer (on the basis of the information I have) is "yes", although I would probably ask the counter-party for the second contract to share the signed document via an online file sharing service (there are many of these, and they are much better for distributing large files than email) or at least to send through signed signature pages.
In theory, a dispute could happen many years down the line, and (for example) email exchanges evidencing the status or terms of the contract might no longer be available or easy to find. For this reason you should always keep copies of ancillary documentation relating to the formation and terms of contracts with the main contract documentation.