If a hydrogen atom accelerates to near the speed of light, the total velocity of the system must not exceed that limit. Even if the electron has no net angular momentum around the proton, as it could be ejected from the atom by photoelectric effect, it must have a discrete velocity around the proton. If this velocity doesn't change then it is possible as the atom accelerates that the sum of this velocity and the velocity of the whole atom exceeds the speed of light. How does the system respond to this eventuality? By decreasing the angular velocity of the electron? But how?
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While your example with the electron has certain problems because of quantum mechanics, your general question has a simple answer: In special relativity, if $B$ is moving with velocity $u$ relative to $A$, and $C$ is moving with velocity $v$ relative to $B$, then the velocity of $C$ relative to $A$ is not simply $u + v$. Instead it is given by the relativistic formula for velocity addition $$ \frac{u + v}{1 + \frac{uv}{c^2}} $$ which you can show is always less than $c$.
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