What and Why Happened
The agreements called for Russia to be the sole heir to the USSR, because, contrary to the official Soviet propaganda, Russia was not just the "first among equals". USSR was a "re-branding" of the Russian Empire.
Thus Russia inherited
- The debt
- The properties abroad (embassies &c)
- Operational control over Nukes
- Treaties, including
- UNSC permanent membership
- nuclear status in the NPT
This was the only natural and generally acceptable solution, similar to the solution taken after the British empire's dissolution. Just like it would have been absurd to make the UK share the UNSC veto power with the British Commonwealth countries (erstwhile colonies), it would have been absurd to make Russia share the UNSC veto power with the Commonwealth of Independent States.
Observe that the above inheritance is a "package" called "Assignment", i.e., one can hardly expect Russia to get some but not all of these items.
Who would want a different solution?
- Do you know of any country who would voluntary dilute its power?
- The former Soviet "republics"?
- They wanted as little association with Russia as possible, in fact, CIS was so loose it might as well not exist.
- The West?
- They were so overjoyed with the end of the Cold War that they bent over backwards to welcome Russia "back into the fold". They were also concerned with non-proliferation, so they wanted all nukes under a single control.
Budapest Memorandum on Security Assurances:
The memorandum included security assurances against threats or use of force against the territorial integrity or political independence of Ukraine as well as those of Belarus and Kazakhstan. As a result Ukraine gave up the world's third largest nuclear weapons stockpile between 1994 and 1996, of which Ukraine had physical though not operational control. The use of the weapons was dependent on Russian controlled electronic Permissive Action Links and the Russian command and control system.
Thus, in theory, Ukraine can claim that, unless US and UK protect it from the Russian aggression, it has the right to re-acquire nuclear weapons (but not claim a share in UNSC membership).
In practice, Ukraine acquiring nukes will be counterproductive, as they will not nuke Moscow anyway and just chill their relationships with the West, while giving a pretext to Russia to invade in earnest.