The breath will have developed its own tempo already because it is associated with the activity of the mind. Simply being aware of the breath is enough for the mind to start to return to peace. As peace begins to be created, you will be able to recognize your deepest aspect, the Awareness.
Meditation is a state of focused awareness. When breath is selected as the object of focus, we simply put our attention on the breath and remain as awareness. In the beginning, when we haven’t recognized the awareness yet, we just focus on the breath. The awareness will begin to emerge as more peace is created in the mind with our practice of focusing on the breath. Then, we can better remain as awareness while we are focusing on the breath.
How deep we can breathe is also related to the health of the body. You don’t need to breathe deep or shallow to be able to meditate. Just focusing on the breath is enough. Meditation is a state of focused awareness. This article accurately and concisely explains all the important points you need to know to be able to practice meditation. It instructs not to try to control the breath during the meditation. This article is explaining meditation aimed to bring us into the experience of our deepest aspects. The goal that is aimed is spiritual enlightenment.
There are other types of breath exercises, where control is applied on the breath. The purpose of these is different. When we maintain the need to control the breath with our mind, this is holding our focus on the mind, but the real purpose of meditation is to transcend the mind. These exercises where breath is controlled in a certain way will also create some change, and since breath is focused on, we will experience more energy, the breath being the flow of energy within ourselves. We can even begin to recognize the state of awareness. But, remaining in the mind level is a limitation. It is wiser to aim to fully transcend the mind and to experience living from our Higher Self state from the beginning. This path does not reject the mind, but also leads to transcending it from the very beginning, creating the highest states of mastery.
Update: Contemplating on this further, and looking it up more, I would like to add that, at certain stages, we might apply breath control/thought control during meditation practice, but even then, the intention must be to prepare ourselves for awareness based meditations, for transcendence. If by mistake we cling to the control aspect of the methodology, thinking that there is something wrong that needs to be fixed about us, it will keep us at the mind level, and in limitation. Two types of meditation practices that use thought control/breath control are Breath Counting Meditation and Ujjayi Breath Meditation. Both will lead to the creation of more peace and recognition of awareness from an active mind state, and then we can practice awareness based meditations with greater concentration. Explore the linked article to learn more about these practices that include thought/breath control and how they prepare us for awareness based meditations, and notice the emphasis on transcendence.
The emphasis of this answer is, rather than fighting with the current state of the mind, or the breath, we must work with them with the intention of transcendence. Following the guidelines of meditation in this answer will lead to a clear understanding of how to do this.