Will any of the CW tv shows be on here anytime soon or in the future
Probably not. While CW decided not to renew their contract with Netflix, Netflix still gets to keep all of the CW shows it had from before the contract expired and all future season of those shows. Any new shows, like Batwoman, will have to be negotiated separately by Netflix. However, Netflix has huge incentive to pursue the rights to stream those shows, so they may offer more money than WB could make by putting the new shows here. And even when the Arrowverse shows eventually leave Netflix, WB has expressed more interest in putting them on the upcoming WB streaming service than on this service.