The primary problem isn't that current software is made for x86. Many systems could be ported if there was a good enough reason. No matter how good Apple's chips are going to be, Apple is very unlikely to share. And they're focusing on a quite narrow part of the industry. For example, they don't seem very interested in making processors for web servers.
Existence of such a processor depends on whether they can sell enough of it. Unlike Apple, they don't have a very popular phone to lean on. And success is not guaranteed. Just because they set out to design an ARM processor doesn't mean it will be able to stand up to the top dog. And it might not be their target at all. They could be targeting mid-range phones, for example, as their entry. Or netbooks and chromebooks (low performance, not high performance). Or they could be looking into server processors (they have in the past IIRC). Intel used to have XScale, so ARM wouldn't be new to them either. I believe Intel still holds license to implement ARM architecture (design their own cores).