Your question falls to an area of physics that is hard to understand - evolution of a system in time according to quantum-theoretic AND relativistic theory of matter.

In non-relativistic theory, one has the Schroedinger equation. The radiation due to system is NOT accounted for in the standard form of this equation. The equation conserves $\langle H \rangle$. The $\psi$ function will change in time but won't get to a point where it resembles atom of Bohr size. It will diffuse and change, but it will get bigger rather than smaller.

In a relativistic theory, there are many approaches and equations, but I do not know if there is some preferred way to model the process you imagine. I have never seen a paper that would try to simulate formation of bound state in time.

On a general level in QFT, one has the action principle for the quantum fields of electron and proton. One can derive equations restricting the fields from this principle and try to extract description of the system in time. But how to set up initial conditions for those fields that correspond to your picture of two particles 1m apart and how to visualize the evolving quantum fields, I do not know.

There is also a so-called Bethe-Salpeter equation that deals with bound states in a way close to QFT, but from the papers on it I've seen I got the impression that it is hard to find solutions and the authors themselves did not think it gives more insight beyond what can be extracted from the non-relativistic equations of Breit type with relativistic corrections (these still do not account for retardation and radiation).

To anyone reading this, if you know of some paper on this, please link it in a comment or post an answer, I would like to read it.

Your question falls into an area of physics that is hard to understand - evolution of a system in time according to quantum-theoretic AND relativistic theory of matter.

In non-relativistic theory, one has the Schroedinger equation. The radiation due to system is NOT accounted for in the standard form of this equation. The equation conserves $\langle H \rangle$. The $\psi$ function will change in time but won't get to a point where it resembles atom of Bohr size. It will diffuse and change, but it will get bigger rather than smaller.

In a relativistic theory, there are many approaches and equations, but I do not know if there is some preferred way to model the process you imagine. I have never seen a paper that would try to simulate formation of bound state in time.

On a general level in QFT, one has the action principle for the quantum fields of electron and proton. One can derive equations restricting the fields from this principle and try to extract description of the system in time. But how to set up initial conditions for those fields that correspond to your picture of two particles 1m apart and how to visualize the evolving quantum fields, I do not know.

There is also a so-called Bethe-Salpeter equation that deals with bound states in a way close to QFT, but from the papers on it I've seen I got the impression that it is hard to find solutions and the authors themselves did not think it gives more insight beyond what can be extracted from the non-relativistic equations of Breit type with relativistic corrections (these still do not account for retardation and radiation).

To anyone reading this, if you know of some paper on this, please link it in a comment or post an answer, I would like to read it.