I just wondered how it stood if we had no contract with the new company
Legally, you almost certainly do have a contract with the new company - they have taken on the rights and responsibilities set out in your original contract.
Check the T&Cs of that contract, there should be an assignment clause covering this, likely stating that (assuming there are no materially significant changes to the terms or services as a result) they can do so without notifying you, or obtaining your permission.
If you don't have a copy, they are required to provide you with one.
and if we were to say that the rate increase was not acceptable to us?
Again, check the terms of your contact. There will be a clause covering increases to membership fees, and whether or not you have an option to reject them via an early/reduced notice of termination.
Finding them unacceptable is only contractually relevant if the increase is manifestly unfair, but you say that's not the case.
There are circumstances in which the term governing increases itself could be considered unfair, but you'd need to post the exact language.
Are there any factors which might convince them to shorten the notice period?
It comes down to the exit/notice terms in your contact...but from the information you've given, it's unlikely.
As long as they're following those terms, the terms are fair and they haven't made significant material changes to them without informing you, there's likely no contractual reason for them to release you early (and gyms generally do not like to set that precedent!)