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There is a subtle issue here but it is not where the OP thinks it is. Any explicitly written integer is obviously "standard" whereas each new integer arising in the ultrapower of $\mathbb N$ is obviously "nonstandard". The subtle issue is that in the particular $\mathbb N$ we are working with, there may be integers (certainly bigger than the explicitly written ones) that look nonstandard from the viewpoint of another set-theoretic universe.

But this does not change the fact that any concrete proof has "standard" length. And since we are accepting ZFC axiomatics for each instance of a universe in Hamkins' multiverse, this has nothing to do with ultrafinitism.

In this publication (see also here) we proved a theorem that each universe in the Hamkins-Gitman "baby model" actually turns out to be a model of a variant of Edward Nelson's Internal Set Theory. Translated into the terms of Robinson's framework, this means that the integers in each instance of a universe in the multiverse contain not only an initial cut including all explicitly specifiable integers, but also a wealth of integers that are infinite according to a "smaller" universe in the multiverse.

One should note that the idea of a "standard $\mathbb N$" being an illusion, as implied in these question and answers, was the main driving force behind Petr Vopěnka's approach to the foundations of mathematics, formulated as his Alternative set theory already in the 1970s, according to what I have read of Vopenka's work.

There is a subtle issue here but it is not where the OP thinks it is. Any explicitly written integer is obviously "standard" whereas each new integer arising in the ultrapower of $\mathbb N$ is obviously "nonstandard". The subtle issue is that in the particular $\mathbb N$ we are working with, there may be integers (certainly bigger than the explicitly written ones) that look nonstandard from the viewpoint of another set-theoretic universe.

But this does not change the fact that any concrete proof has "standard" length. And since we are accepting ZFC axiomatics for each instance of a universe in Hamkins' multiverse, this has nothing to do with ultrafinitism.

In this publication (see also here) we proved a theorem that each universe in the Hamkins-Gitman "toy model" actually turns out to be a model of a variant of Edward Nelson's Internal Set Theory. Translated into the terms of Robinson's framework, this means that the integers in each instance of a universe in the multiverse contain not only an initial cut including all explicitly specifiable integers, but also a wealth of integers that are infinite according to a "smaller" universe in the multiverse.

One should note that the idea of a "standard $\mathbb N$" being an illusion, as implied in these question and answers, was the main driving force behind Petr Vopěnka's approach to the foundations of mathematics, formulated as his Alternative set theory already in the 1970s, according to what I have read of Vopenka's work.

There is a subtle issue here but it is not where the OP thinks it is. Any explicitly written integer is obviously "standard" whereas each new integer arising in the ultrapower of $\mathbb N$ is obviously "nonstandard". The subtle issue is that in the particular $\mathbb N$ we are working with, there may be integers (certainly bigger than the explicitly written ones) that look nonstandard from the viewpoint of another set-theoretic universe.

But this does not change the fact that any concrete proof has "standard" length. And since we are accepting ZFC axiomatics for each instance of a universe in Hamkins' multiverse, this has nothing to do with ultrafinitism.

In this publication (see also here) we proved a theorem that each universe in the Hamkins-Gitman "baby model" actually turns out to be a model of a variant of Edward Nelson's Internal Set Theory. Translated into the terms of Robinson's framework, this means that the integers in each instance of a universe in the multiverse contain not only an initial cut including all explicitly specifiable integers, but also a wealth of integers that are infinite according to a "smaller" universe in the multiverse.

There is a subtle issue here but it is not where the OP thinks it is. Any explicitly written integer is obviously "standard" whereas each new integer arising in the ultrapower of $\mathbb N$ is obviously "nonstandard". The subtle issue is that in the particular $\mathbb N$ we are working with, there may be integers (certainly bigger than the explicitly written ones) that look nonstandard from the viewpoint of another set-theoretic universe.

But this does not change the fact that any concrete proof has "standard" length. And since we are accepting ZFC axiomatics for each instance of a universe in Hamkins' multiverse, this has nothing to do with ultrafinitism.

In this publication (see also here) we proved a theorem that each universe in the Hamkins-Gitman "baby model" actually turns out to be a model of a variant of Edward Nelson's Internal Set Theory. Translated into the terms of Robinson's framework, this means that the integers in each instance of a universe in the multiverse contain not only an initial cut including all explicitly specifiable integers, but also a wealth of integers that are infinite according to a "smaller" universe in the multiverse.

One should note that the idea of a "standard $\mathbb N$" being an illusion, as implied in these question and answers, was the main driving force behind Petr Vopěnka's approach to the foundations of mathematics, formulated as his Alternative set theory already in the 1970s, according to what I have read of Vopenka's work.

There is a subtle issue here but it is not where the OP thinks it is. Any explicitly written integer is obviously "standard" whereas each new integer arising in the ultrapower of $\mathbb N$ is obviously "nonstandard". The subtle issue is that in the particular $\mathbb N$ we are working with, there may be integers (certainly bigger than the explicitly written ones) that look nonstandard from the viewpoint of another set-theoretic universe.

But this does not change the fact that any concrete proof has "standard" length. And since we are accepting ZFC axiomatics for each instance of a universe in Hamkins' multiverse, this has nothing to do with ultrafinitism.

There is a subtle issue here but it is not where the OP thinks it is. Any explicitly written integer is obviously "standard" whereas each new integer arising in the ultrapower of $\mathbb N$ is obviously "nonstandard". The subtle issue is that in the particular $\mathbb N$ we are working with, there may be integers (certainly bigger than the explicitly written ones) that look nonstandard from the viewpoint of another set-theoretic universe.

But this does not change the fact that any concrete proof has "standard" length. And since we are accepting ZFC axiomatics for each instance of a universe in Hamkins' multiverse, this has nothing to do with ultrafinitism.

In this publication (see also here) we proved a theorem that each universe in the Hamkins-Gitman "baby model" actually turns out to be a model of a variant of Edward Nelson's Internal Set Theory. Translated into the terms of Robinson's framework, this means that the integers in each instance of a universe in the multiverse contain not only an initial cut including all explicitly specifiable integers, but also a wealth of integers that are infinite according to a "smaller" universe in the multiverse.