First, note that all what is fixed about $|\psi'\rangle=U|\psi\rangle$ and $|\phi'\rangle=U|\phi\rangle$ is their scalar product -- any other property can be changed by choosing a suitable $U$.
We are thus considering arbitrary $|\psi'\rangle$ and $|\phi'\rangle$ with a fixed scalar product $\langle \psi'|\phi'\rangle=\alpha$, which satisfy property 2., and we want to prove they must be equal.
To this end, choose $|\psi'\rangle=|0\rangle|0\rangle$, and
$|\phi'\rangle=\alpha|0\rangle|0\rangle+\sqrt{1-|\alpha|^2}|0\rangle|1\rangle$. Then, $\langle \psi'|\phi'\rangle=\alpha$, yet, $|\phi'\rangle$ does not have the same Schmidt basis, in contradiction with property 2 -- unless $|\alpha|=1$, which implies that $|\psi'\rangle\propto|\phi'\rangle$.
Your claim already fails in the simplest non-trivial case, namely for three qubits.
This follows from a (the?) classic result in the theory of multipartite entanglement -- Three qubits can be entangled in two inequivalent ways by Dür, Vidal, and Cirac.
They show that for three qubits, there are two classes of genuine tripartite entangled states, the W class and the GHZ class. Given states $\lvert \phi_\mathrm{W}\rangle$ and $\lvert \phi_\mathrm{GHZ}\rangle$, from the two classes, it is impossible to convert between them using SLOCC, i.e., it is impossible to write
$$
\lvert\phi_\mathrm{W}\rangle\propto (A\otimes B\otimes C)\lvert\phi_\mathrm{GHZ}\rangle
$$
(or vice versa) for any $A$, $B$, and $C$. In particular, this rules out the possibility of unitaries doing the job, as you are asking.
What remains is to show that there are states $\lvert\phi_\mathrm{W}\rangle$ and $\lvert \phi_\mathrm{GHZ}\rangle$ with identical Schmidt coefficients in every bipartition. This is equivalent to demanding that their single-qubit reduced states have the same spectra.
Using the results of the paper of Dür, Vidal, and Cirac, it is now easy to find such states (and indeed, they really should exist -- it would be rather disappointing if the two classes were distinguished by the spectra of their reduced density matrices). For instance, you can choose
$$
\lvert \phi_\mathrm{W}\rangle =
\sqrt{\gamma}\lvert 001\rangle +
\sqrt{\gamma}\lvert 010\rangle +
\sqrt{\gamma}\lvert 100\rangle +
\sqrt{1-3\gamma}\lvert 000\rangle
$$
(cf. Eq. (20) in the paper), and
$$
\lvert \phi_\mathrm{GHZ}\rangle
\propto
\lvert 0\rangle \lvert 0\rangle \lvert 0\rangle +
\lvert \theta\rangle\lvert \theta\rangle\lvert \theta\rangle
$$
(cf. Eq. (15)),
with $\lvert\theta\rangle=\sqrt{\mu}\lvert 0\rangle + \sqrt{1-\mu}\lvert 1\rangle$.
You can now easily check that for the spectra $(\lambda,1-\lambda)$ ($\lambda\le1/2$) of the single-qubit reduced states (which are all equal by symmetry)
- for $\lvert\phi_\mathrm{W}\rangle$, all values $0<\lambda\le1/3$ can be obtained by varying $0<\gamma\le1/3$
- for $\lvert\phi_\mathrm{GHZ}\rangle$, all values $0<\lambda\le1/2$ can be obtained by varying $0\le\mu< 1$.
Thus, you can easily find values $\mu$ and $\gamma$ where the reduced states have identical spectra, and thus the Schmidt spectra in all bipartitions are equal, yet the states cannnot be converted into each other by local unitaries (or even SLOCC).
Best Answer
No.
To this end, consider $$ \lvert\phi\rangle = a\lvert0\rangle\lvert0\rangle + b \lvert1\rangle\lvert1\rangle\ , $$ and $$ \lvert\psi\rangle = a\lvert+\rangle\lvert+\rangle + b \lvert-\rangle\lvert-\rangle\ , $$ where $a=\sqrt{\tfrac12-\varepsilon}$, $b=\sqrt{\tfrac12+\varepsilon}$ [and with $\lvert \pm\rangle = \tfrac12(\lvert0\rangle\pm\lvert1\rangle)$].
$\lvert\phi\rangle$ and $\lvert\psi\rangle$ are in their Schmidt decomposition, and it is unique (as long as $\varepsilon\ne 0$). Moreover, $$ \|\lvert\phi\rangle\langle\phi\rvert-\lvert\phi\rangle\langle\phi\rvert\|_p \to 0 $$ as $\varepsilon\to 0$.
Yet, their Schmidt vectors do not become close to each other; in fact, they are completely independent of $\varepsilon$.
Thus, the only way in which this can be made to work is if you insist that you are sufficiently far (as comapred to $\varepsilon$) from a state with degenerate Schmidt coefficients.