You can mimic a radio button really easily using just barebones RN. Here’s one simple implementation which I use. Tweak size, colors etc. as you like. It looks like this (with a different tint, and some text). Add TouchableOpacity on top to turn it into a button that does something.

function RadioButton(props) {
return (
<View style={[{
height: 24,
width: 24,
borderRadius: 12,
borderWidth: 2,
borderColor: '#000',
alignItems: 'center',
justifyContent: 'center',
}, props.style]}>
{
props.selected ?
<View style={{
height: 12,
width: 12,
borderRadius: 6,
backgroundColor: '#000',
}}/>
: null
}
</View>
);
}