If we assume that protons don't decay, then will all matter ultimately decay into Iron-56 or into nickel-62?

The binding energy curve, again in wikipedia, shows iron as the one with the smallest binding energy per nucleon. Though in the table, the following is stated:

56Fe has the lowest nucleon-specific mass of the four nuclides listed in this table, but this does not imply it is the strongest bound atom per hadron, unless the choice of beginning hadrons is completely free. Iron releases the largest energy if any 56 nucleons are allowed to build a nuclide—changing one to another if necessary, The highest binding energy per hadron, with the hadrons starting as the same number of protons Z and total nucleons A as in the bound nucleus, is 62Ni. Thus, the true absolute value of the total binding energy of a nucleus depends on what we are allowed to construct the nucleus out of. If all nuclei of mass number A were to be allowed to be constructed of A neutrons, then Fe-56 would release the most energy per nucleon, since it has a larger fraction of protons than Ni-62. However, if nucleons are required to be constructed of only the same number of protons and neutrons that they contain, then nickel-62 is the most tightly bound nucleus, per nucleon.

One sees that there is a leeway when constructing models in end of the universe scenaria. There is so much speculation in the time lines. Observation tells us that Ni-62 is not abundant, while Fe is. It seems that in the sequence of supernova explosions iron wins out; according to the quote above, this would mean that it is the number of nucleons that is important and the charges statistically arrange themselves.

Anyway, in a continuously expanding universe with a stable proton it is hard to see how the expanding gases of Helium and Hydrogen can tunnel into anything as they expand so that "all matter" would end up as Fe or Ni atoms .

Not to worry, the proton will decay according to most current models of particle physics anyway.

To get Ni62 requires the production of nuclei heavier than Fe56. The problem is that these iron-peak elements are mostly produced in rapid nucleosynthesis reactions in the centres of stars (either massive stars, or in type Ia supernovae). The iron-peak elements are produced in a nuclear statistical equilibrium by burning Silicon. Rapid alpha capture reactions compete with photodisintegration and can successfully produce nuclei up to and including Ni56. To produce more stable isotopes with $N>Z$ then requires weak flavour changes to turn protons into neutrons.

Beyond Ni56 this there is a blockage to further alpha-capture. Zn60 and Ge64 have a lower binding energy and furthermore, the coulomb barrier to alpha capture is higher. At the higher temperatures required to drive these fusion reactions then photodisintegration is capable of breaking down the nuclei more rapidly than they can form.

Thus, even though it has (marginally) more binding energy per nucleon, the path to forming Ni62 in the universe is not favoured. Instead we get lots of Fe56 which is made by two electron captures onto Ni56 in high density regions or by positron decay via Co56 (for example in the ejected envelopes of supernovae).