Changing the hypercharge of the Higgs field

I'm trying to solve a question about the Higgs mechanism:

If the hypercharge of the Higgs were to be $Y=1$ (Changing from $Y=1/2$ ), what would be the photon of this world? Meaning, how can I describe the field? Moreover, what are the charges of the quarks and the leptons? And what is the coupling constant of the electromagnetic force?

I think the new formula for the charge is $Q=Y+u-d$, but I don't know how, or if, the field $A_\mu$ will change.

I gather you are using the (sensible!) "minority usage convention" for the Weak Hypercharge, $Q=T_3+Y$, so, then, half of what appears on this WP article table. This is the simplest and most tasteful one, anyway. (But not for long...)

You wish to contemplate a hypothetical world in which the Higgs hypercharge grows from 1/2 to 1, so it doubles. Since the hypercharge is the quantity least connected to observations, let us take g to stay the same. It is a little like taking your bicycle apart and putting it back together with double its length, to see how it works.

The starting point is the Higgs mechanism, "the first job" of the Higgs. The crucial piece in this mechanism which enters in the gauge field mass matrix to be diagonalized is the hypercharge U(1) gauge field $B_\mu$ coupling to the Higgs. Was $$D_\mu H= \partial_\mu H -ig {\mathbf W}_\mu \cdot \frac{\mathbf \tau}{2} H -i \frac{g'}{2} B_\mu H ,$$ now going to $$... -i g' B_\mu H ,$$ keeping everything else the same. We can then use all the formulas of the standard model the same, substituting $g''\equiv 2g'$ for $g'$ in all the expressions.

So, for example, the Weinberg angle will now increase from about 30 degrees to about 49 degrees, as $$g''/g=\tan \theta_W''= 2 \tan \theta_W \approx 2/\sqrt{3}.$$ As a result, there is more mixing, and the Z grows even heavier than the W, and the photon becomes less hyperchargy $B_\mu$ and more $W^3_\mu$-like, $$A_\mu = \sin \theta''_W ~ W_\mu ^3+ \cos \theta''_W B_\mu~.$$ The electric charge $e=g \sin \theta''_W$ will thus go up by a factor of 1.5. You can further see the coupling ZWW decreases, etc...

In essence, we have transported ourselves to 1972, before neutral currents and the measurement of the Weinberg angle.

Now, however, $Q=T_3 +1$ won't do after the $1\mapsto 2$ transition, anymore, as the Higgs doublet needs to be anchored to have charges $0,\pm 1$, for the goldstons to be eaten right by the W triplet! ... and the leftover H to stay neutral.

We must then adjust this to $Q=T_3+Y/2$, and for U(1) invariance, carry this over to the Yukawa couplings, the second job of the Higgs, so they all conserve weak isospin, hypercharge, and charge, as before. You must then adjust your hypercharge table for fermions, to actually yield the present charges, so $Y(e^-_L)=-1=Y(\nu_L)$, $Y(e^-_R)=-2$, $Y(u_L)=1/3 =Y(d_L)$, etc...

So charges and their assignments have stayed the same, but the hypercharges of the fermions have adjusted to the new charge-isospin-hypercharge relation: the hypercharge is now twice the average charge of the isomultiplet. I understand this is not what you had in mind, but I gather you see the logic of it now.