Building native app experience

Some years ago I found myself in a similar situation. I had mainly worked with web UIs as a developer and interaction designer (IxD) but took on being a UX lead over web, iOS native, and Android native projects.
From a UX perspective this the approach I took:

• Treat each platform as a first class citizen, study Apple's and Google Material design guidelines equally.
• Understand the major differences between platforms and how they affect UX. Navigation, notifications, and native components have different interaction patterns.
• Look for UX designers/engineers doing solid work in mobile UI design, not just Dribble designers doing mocks. Nick Babich (babich.biz) has many good mobile focused articles as well as solid UX articles.
• Find some native developers to work with or get feedback from on your designs. I learned a ton from the diehard iOS and Android devs I worked with. There's alot to keep up on across platforms, and having specialists help you is a huge boon.
• You can wireframe/sketch one design, but prototype for both. Testing and designs are just different in each, with some different paradigms. Especially navigation.
• When I hire for projects around native mobile, I look for designers that aren't just focused on one platform with their portfolio. If they only have iOS designs that's a red flag. If they show a design in multiple platforms and using the platform paradigms, or explain why they diverged from a paradigm, that is a major bonus.
• If you can, get access to a phone from each platform. I bought used phones and dug into major app designs for each. It's really interesting looking at the same app on different platforms. Analyze the differences in major apps like Evernote, Google's, etc.
• Then pay attention to when you need to break away from the design guidelines and not follow them blindly. Some of the features in each platform aren't used by the average user and your user base will be different. 3D gestures in Apple and the notification tray in Android never tested well when I did studies with my target users, so we avoided depending solely on those and other native features in our apps. Consumers vs enterprise users have different needs. Our users also worked in teams side-by-side with mixed devices, so iOS users had to understand the Android version of our platform and vice versa.
• Also, some mobile apps are so popular they become a design convention. Love or hate them, pay attention to the designs of apps with the highest usage. Youtube, Gmail and Facebook fall into this category. In UX, convention and consistency are king.

Hope that helps!
Wes