Do banks care how you saved for a mortgage?
No, not really. They will be more comfortable if you have cash savings rather than a share in some dodgy fine wine scheme (but you'll cash in any non-cash investments before buying anyway) and they will want to know that you've not borrowed to fund the deposit. Beyond this, I can't see it mattering much to the banks whether you save in one account or two. If one of you likes spending, it may be better for the other to organise the savings.
Nationwide are one of the better lenders but you need to look at the whole scheme and it's best not to limit your options. If you like any particular lender's mortgage products, saving with them will help a little with the application but it wont be a major factor.
Yes, the tax benefits of ISA usually make it worthwhile to use the full limit before saving elsewhere. Whether you go for both types of ISA will depend on your attitude to risk.