A Higgs-less Universe would have no Higgs field and no Higgs mechanism (surprise surprise). There is one huge consequence, namely the fact that in that scenario no electroweak symmetry breaking happens.
Let us concider the most simplistic scenario where the Standard Model (SM) Lagrangian contains no Higgs sector. We will take the gauge group to be the SM one, that is
\begin{equation}
SU(3)_{c} \times SU(2)_L \times U(1)_Y,
\end{equation}
and we will assume that gauge couplings are the same and likewise is the fermion sector of the theory. We stress again that only the Higgs sector is completely absent. Then, such an elimination would have absolutely no effect to the QCD part of the theory. QCD would still be a confining theory for color-triplet quarks and color-octet gluons into color-singlet hadrons, i.e. mesons and baryons. Now, if electroweak symmetry remains unbrocken, the what in the SM is asymptotically free $SU(2)_L$ force, would become confining too. This imples the existence of confined weak isospin particles into weak isospin singlets. Remember though that QCD spontaneously breaks the chiral symmetry of the massless $u$ and $d$ quarks, at some scale $\Lambda_{\text{QCD}}$, as
\begin{equation}
SU(2)_L \times SU(2)_R \to SU(2)_V,
\end{equation}
resulting in a condensate capable of not making the electroweak symmetry apparent even when the Higgs mechanism is absent. Three Goldstone bosons would appear. Then, as mentioned, the $\langle \bar{q}q \rangle = \langle \bar{q}_Lq_R + \bar{q}_Rq_L \rangle$ would make the left- and right-handed quarks transform differently under $SU(2)_L \times U(1)_Y$. Then, because of $I_3$ and $Y_W$ numbers of the left- and right-handed quarks the condensate would transform as a weak isodoublet with weak hypercharge $Y_W = |1|$ and this would end up breaking $SU(2)_L \times U(1)_Y \to U(1)_{EM}$. Then the massless pions of this theory would ``dissapear" from the spectrum as they would become the longitudinal components of the weak gauge bosons. The spectrum would also contain a gauge field $A_{\mu}$, the photon. Since the weak bosons acquired mass the $SU(2)_L$ would not be confining. Still, the quarks and the leptons, would remain massless since in QCD-like electroweak symmetry breaking there does not exist a Yukawa couplings generating mechanism and since there is no Higgs boson the interactions of this theory would become strongly coupled.
Now, note that altough the pions would be absent, the spectrum of this Higg-less SM would consist of light hadrons and despite the fact that the spectrum would would be very similar to the one of the SM there would be very significant, as we will see, differences. In the Higgs-less SM, as we mentioned before, the quark masses would be zero, so, for example, there would be no mass hierarchy or mass difference between the $u$ and $d$ quarks able to overcome the electromagnetic self-energy of the proton which leads to the latter decaying to the neutron in the SM. Then the bets would be on the $\gamma$ and $Z^0$ mass modulations and it would turn out that the difference in the masses of the $u$ and $d$ quarks would be $\mathcal{O}(1)$ MeV. This would have dramatic effects! The $\beta$-decay would go "crazy". In the SM a free neutron decays through $n \to pe^{-}\bar{\nu}_e$ with $\tau \approx$ 15 minutes. If the sign in the mass difference changed we could have $p \to ne^{+}\nu_e$ with $\tau \approx \mathcal{O}(10^{-12})$s for weak bosons near $\Lambda_{\text{QCD}}$. This implies that no hydrogen atoms could be formed since the protons would decay! No hydrogen also would mean that in this Higgs-less SM that the lightest nucleus would be a neutron. Overall the whole theory of Big Bang nucleosynthesis would be different since atoms would be impossible to form. Going a step further though let us assume that some elements could somehow form. They would be completely unfamiliar and different to the ones we know and of course unstable.
Finally, another remark is that the electron would also be massless as we have already realized. This would mean that the Bohr radius, that in the SM is of $\mathcal{O}(10^{-9})$m, would be infinite making matter not well defined in the sense that it would be without integrity. Then we realize that there could not be compact atoms and as a result bonds would also make no sense. There would be no chemistry, no biology and as a result no life and all that in an unstable vacuum that would continuously produce $e^{+}e^{-}$ pairs.
We are glad that the Higgs field exists and the electroweak symmetry breaking happened to allow us to be here although we cannot say that there are not alternatives that would create a Universe like ours without the nessecity of a Higgs mechanism. I think the above are quite accurate but maybe a phenomenologist (unlike me) can provide a better explanation.
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