Today's security is based on $RSA$ and $DLOG$ primarily. If both go down then it is hard to imagine a scenario where internet can be salvaged quickly.
Assume $RSA$ with $n$ bit prime factors and $DLOG$ with $n$ bit prime modulus have a (possibly practially realizable (Shor's algorithm still has challenges in implementation) quauntum) algorithm with $\leq(20n)^{2.5}$ time (picked an exponent on par with today's linear programming complexity exponent and $20$ was a reasonable small number). Then if we take $10^{25}$ arithmetic operations as ultimate limit of realizable complexity then if $n\leq 5\times 10^8$ holds we can break it ($(20n)^{2.5}\leq10^{25}\iff(20n)^{}\leq10^{10}\iff2n\leq10^9$).
How large can $RSA$ and $DLOG$ keys be before the standard gives up?
Is it reasonable to assume $10^9$ bit keys are possible with next generation internet?