Surely the base you learn with is arbitrary?
I would argue that if you raised your child in total isolation, meaning that it wouldn't ever encounter the base 10 decimal system, this would hold true. The common assumption why we prefer base 10 is because we have ten fingers, but this never convinced me that base 10 is more "natural" than, say, base 6 (we could hold bigger numbers if we counted using a two-digit system where each hand represented one digit in base 6). Also, not everyone learns to count using their fingers, so it should be possible to teach a child maths in an other base and have it be just as proficient.
should my child become just as good as anyone else doing basic multiplication and other simple maths if they only learnt in base 16
I don't think so. Learning usually isn't an isolated activity. Kids encounter counting long before they go to school, in rhymes, songs, at birthday parties and so on. There are a ton of every-day interactions where mathematical knowledge gets taught to very small children. They hear other people counting balloons or cars or stars in the sky. If your kid expects to hear "A" but people call that number "ten", that's got to lead to a serious amount of confusion. Also, when your kid tells his/her peers that his older brother just turned "C", they're not going to understand and will think he's either very weird or plain stupid (can't even count right). Kids don't want to be different; they want to fit in and be accepted. You make that harder if you teach them to use another numbering system as the default.
Also, since the world is running on base 10, we tend to do things in units of 10 - so the number 10 will appear much more often in the real world than the number 16 (things will often be organized in groups of 10, for example). Multiplication and division are very easy in these cases in base 10, much harder in base 16.
Plus, of course, your child will have to spend the rest of his/her life converting numbers from base 10 to base 16 (and vice versa) just to be able to function in our society. That's a serious amount of work that will slow down your child, even if it can do the actual math just as fast as everyone else (just in base 16).
So, to summarize, if this is a hypothetical question where a child never gets in touch with the real world, then yes, it will most likley perform just as well in base 16. If this is something you actually consider doing, then just don't. You'd make your child's life more complicated without cause.
What I think would happen if you actually tried this is that your child would start out counting in base 16 and then switch over to base 10 when he/she noticed everybody else was doing it in base 10, just to get rid of doing the extra work of converting everything he hears and says. This would happen fairly early (before he/she was ready to learn how to multiply and divide, with the child knowing maybe numbers 1-20 dec). He'd be "bilingual", talking in base 16 with you to make you happy, and using base 10 everywhere else. If you insisted on him learning to multiply and divide in base 16, he'd probably slow down in comparison to everybody else because he'd have to learn to do the same thing in two numbering bases at roughly the same time. There are studies showing that kids who grow up bilingual tend to be less proficient in both languages than monolingual kids, which makes sense to me because they spend their time learning two instead of just one language, which is harder. I'd assume the same is true for doing math in two bases in parallel.
