Anything in the realm of employer-employee relations is governed solely by what is legal. What is "customary" does not matter, because, should the situation arise, the company would not hesitate to throw "customary" out the window if it is in their best business interest. If the company wanted you to give them 2 weeks notice, they would have put it in the contract. If they didn't put it in the contract, that means they don't need it.
It is the job of the employer, when they prepare the contract, to put in any and all provisions to protect themselves. If they are concerned about continuity vis a vis notice periods, it's their job to put that in your contract, that you need to give 2 weeks notice. If they didn't put that in the contract, it means they don't care about you giving 2 weeks notice and they are prepared to replace you immediately, or be short-staffed for a period. It's not your job, as an exiting employee, to ensure continuity of the company which you are leaving, beyond whatever you are required to do vis-a-vis preparing documentation and offboarding and so on (and even that is mostly voluntary, or "customary" if you prefer).
All of this is a long way of saying: When the new company asks you how soon you are available, if your contract does not stipulate a notice period, the answer is "when would you like me to start?". That's how soon you are available. Two weeks notice or not, your priority can't be both to your old company and to your new company. Look forward, not back; give the new company every reason to give you an offer and no reasons (as few as possible) to not do so.