I can't see how to make a noble bastard child prodigy. The noble born gets one trait that he has to spend on Mark of Privilege. I looked that up to make sure I wasn't crazy. Then I go to bastard. You get two traits but you have to spend one on bastard. So that leaves you with 1 trait point and Child Prodigy costs 2 and that's all the lifepaths you get.
That's right, you can't afford the Child Prodigy trait if you choose the Born Noble → Bastard set of lifepaths.
As the game says, choices have consequences, and choosing to burn up a noble bastard means not being able to choose Child Prodigy and its awesome bonuses. If you want Child Prodigy with a noble, consider Born Noble → Young Lady: she has two trait points to spare.
It's also easy to get Child Prodigy with Born LPs that aren't from the Noble Setting, such as Peasant Born → anything, or City Born → Runner, for just two lifepath sets that could produce a real Child Prodigy.
Basically, the awesome mental abilities granted by the Child Prodigy trait simply isn't available to anyone who is already born with a silver spoon in their mouth and a claim to power in their future. That's a deliberate detail of the implied setting of the Burning Wheel.
Put in game-balance terms, Child Prodigy is just that good, and the game forces you to make sacrifices if you want it badly enough. You have to really want it and build everything else around it to get access to it. If a player is unwilling to do that, they don't get the payoff of Child Prodigy's bonuses.
You are correct: Born Noble → Bastard leaves you with only 1 trait point to allocate freely, whereas Child Prodigy costs two. So you can't literally make a Child Prodigy Born Noble → Bastard character in so many words.
Burning Wheel can be a bit flexible about how exactly you represent a character concept, though. Here are some alternative ways to create a similar character:
City Born → (leads to Noble) Bastard: Representing a character with just a little bit less status than the full Noble Born bastard. Perhaps you were only acknowledged later in life, or you've simply ended up being raised more like a courtier or functionary than a "spare heir" — but as you've matured, you've come into your own socially and now whether or not you were truly acknowledged is a rather moot point. Mechanically, being a 19-year-old puts you at the prime of your life in terms of stats.
Gifted Child (from Magic Burner; use the Noble setting) → Bastard: Gifted Child is an LP designed to give you access to the Gifted (magic-user) trait, but you don't have to take it. Instead, play up your Child Prodigy status as an uncanny, perhaps supernatural-seeming talent — you are Misunderstood (mandatory trait), after all.
Maybe you don't need the Child Prodigy trait? Born Noble → Bastard → Young Lady, Student, or Page is a perfectly good three-lifepath character. For your Jon Snow types &c.
Maybe you don't need the Bastard trait per se? It's a complex Die Trait designed to represent the illegitimate (but, implicitly, acknowledged) child of a "noble of rank." I could imagine, instead, creating a character with a Character Trait like "Unrecognized Bastard" and a belief about attaining recognition in some way: you don't have any of the social capital of a capital-B Bastard, just some tenuous thread of potential (that might as well be made up, as far as most others are concerned).
Just one caveat here, especially towards the last example: while the lifepath system isn't a perfect delicate engine that you're never allowed to game or tweak, consider that the reason you're sorta bumping into these restrictions in the first place (not enough trait points for all these different traits, Bastard being a 4-point Die Trait unless you take it from the lifepath) is that the character you have in mind might be kind of "busy:" being a noble at all, being a noble bastard, and being amazingly talented are all kind of major "shticks" in their own right. Part of the logic of the trait system is to make you pay more for qualities that have major narrative impact, even if they're traits like "Blind." I certainly wouldn't treat it as a hard block, just a small red flag to let you know to be extra careful about running the idea past your play group.