Nuclear testing was largely phased out by the Comprehensive Nuclear-Test-Ban Treaty in 1996, with the exception of non-signatories like India, Pakistan (both last testing weapons in 1998) and North Korea (tests during the 2010s).

I see two arguments in favor of restarting some limited nuclear tests, and a couple of weak counter-arguments.

Reinstating such tests would include public announcement of their time & place, with penalties for unannounced tests.

Arguments:

1. Testing of warning systems: Early warning systems malfunctioning present a large risk of normal events being mistaken as attacks, with correspoding nuclear counter-attacks. Reading through the FLI Close Calls Timeline, I identify 7/28 events as being clear examples of malfunctioning early detection systems as causes for close calls (e.g. "Lost Contact with 50 Missiles", "Six Mis-Routed Nukes", "Soviets Misinterpret US Nuclear War Games" (which would be much easier under a setup where announced military testing was expected and acceptable), and "False Alarm During DEFCON 3") and 6/28 more as being at least linked to detection systems malfunctioning ("Norwegian Rocket Mistaken for ICBM", "Soviet Union Detects Incoming Missiles", "Faulty Chip Signals Soviet Attack", "Power Failure Mistaken for Nuclear Blasts", "Missiles over Georgia"). However, there is at least one case in which a test was perceived as an attack ("Soviet Missile Headed Toward US?") — could this one have been avoided if announcing nuclear tests was common? Additionally, some false alarms (e.g. "Tampa is getting nuked") result from participants mistaking training tapes for real attacks, which would not happen as much in the case of real-world tests. Having real-world nuclear tests occurring regularly might be a time to test early detection systems and thereby guard against false positives by increasing the specificity of early detection systems, thereby lowering the risk of accidental escalation based on faulty information. My best guess is that operators of those early warning systems are very uncertain about their reliability, and would become significantly more relaxed if they had real-world information to calibrate on.
2. Reducing redundancy: Similarly, unless covert nuclear testing is occurring (which I consider unlikely${}_{10\%}$, given the related intensive monitoring), I expect there to be significant uncertainty in the military commands of nuclear states about the reliability of those weapons, with conservatism about the reliability of those weapons leading to larger stockpiles ("If we don't know whether our weapons work, it's probably better to make more & different ones."). This uncertainty probably leads to an effective number of nuclear weapons that is larger than strictly necessary for military purposes, and has large negative externalities—the weapons would probably still be used in a nuclear conflict. Nuclear testing could reduce this uncertainty about the reliability of the weapons, making reductions in stockpiles more realistic. If I imagine having to rely on weapons that haven't been tested in more than 30 years, I'd also want to err on having more of those weapons rather than less.

There are some counterarguments to restarting the testing:

1. Tests as cover for attack: More tests would mean that countries could mistake tests as attacks, or tests be used as a plausibly deniable scenario in which to strike first (perhaps even masking the attack as an accident). This risk could be strongly reduced by requiring public announcements of the tests e.g. a month ahead.
2. Environmental impact: Wikipedia states that nuclear fallout has lead to an excess of ~11,000 deaths due to nuclear radiation. Resuming (restricted) nuclear tests would likely have similarly high externalities—could these perhaps be mitigated in some way, such as testing in very remote (internationally shared?) locations, such as the Antarctic?
3. Making more powerful weapons: Nuclear tests might enable the development of even more powerful nuclear weapons. I'm not sure how bottle-necked the development of bombs with higher yields is on experimental data: the reason I know about the limit of the size of nuclear weapons is that "there is no target big enough to be hit by them". If this is not true, then more destructive nuclear weapons developed using experimental data would be negative for the world. (Relatedly, I'm not sure whether, if nuclear testing allow for weapons with higher precision, that would be good or bad for the world: nuclear strikes would have fewer externalities, but leaders might also be more willing to use nuclear strikes due to lower civilian casualties (and thereby increasing the risk of nuclear escalation)).