To make this simpler for illustrating the basic relationships, let's assume that the fraction of the population with the gene mutation is sufficiently small that including or excluding those individuals doesn't make an appreciable difference in the overall population baseline probability of developing breast cancer. That's not a bad assumption for the germ-line mutation that you posit. As you mention a hazard ratio, let's assume that a proportional hazard (PH) assumption holds with respect to having gene mutation $g$, with hazard ratio $\text{HR}_g$.

You have a function $F_0(t)$ for the baseline cumulative probability of breast cancer in the population over time $t$.* It's simpler to work with its associated survival function, $S_0(t) = 1- F_0(t)$, and the corresponding cumulative hazard function $\Lambda_0(t) = -\log S_0(t)$.

Then under PH the cumulative hazard for those with the gene is:

$$\Lambda(t|g) = \Lambda_0(t) \text{HR}_g .$$

That comes directly from PH and the definition of a hazard ratio. Plugging back into the relationship between cumulative hazard and survival:

$$S(t|g) = \exp(-\Lambda(t|g))= S_0(t) ^{\text{HR}_g},$$

taking baseline fractional survival to the power of the HR. Subtract that from 1 to get the cumulative probability of the event through that time.

Odds ratios or risk ratios are most generally defined with respect to some particular survival time $t$. You have to be particularly careful about use of the word "risk" and the abbreviation "RR." For example, Rodríguez calls the hazard ratio a "relative risk" (Section 2.3), which one might choose to abbreviate "RR." But if you think in terms of a risk (probability) ratio for developing breast cancer through some time $t$, also potentially abbreviated "RR," you see a more complicated relationship:

$$ \text{probability/risk ratio}(t)=\frac{F(t|g)}{F_0(t)} =\frac{1-S_0(t) ^{\text{HR}_g}}{1-S_0(t)} .$$

The odds of developing breast cancer by time $t$ is $F(t)/(1-F(t))=F(t)/S(t)$. So the odds ratio between those with $g$ and the population baseline is:

$$\text{odds ratio} (t)= \frac{1-S_0(t) ^{\text{HR}_g}}{S_0(t)^{\text{HR}_g}} \frac{S_0(t)}{F_0(t)}.$$

Those won't in general be constant over time under the PH assumption.

*In practice this isn't a proper probability distribution, as not all individuals will develop breast cancer. This might be modeled better as a cure or, particularly as mutations in genes like ATM increase the probabilities of many types of cancer, a competing-risks model. We'll set those issues aside for this answer. Rodríguez discusses some implications of improper distributions in these course notes on survival models. There's also a serious question whether the PH assumption will hold in this situation.