This claim is misleading and could even be opinion-based. There are many complex factors influencing which country has the most "modernized" nuclear weapons.
Russia and the US are the two top competitors for the most modernized nuclear weapons. Russia’s Nuclear Weapons are focused on the delivery vehicles which are the SS-27 Mod 2 (Yars) ICBM and the SS-X9 (Sarmat) heavy ICBM. Russia hasn't shared information regarding nuclear strategies with the US. However, Putin continues to try to instill fear by adding that they are testing newly developed weapons.
Whereas the US focus for modernization of its weapons is within the Airforce on the LGM-35MA Sentinel (ICBM), continually working on the B-2 Fleet, the redesign of B-52s, and the Navy plans to launch the "Columbia-Class" of missiles.
Looking at the stockpiles of nuclear weapons Russia has more with the US closely behind. This may influence people to believe that Russia is ahead in the nuclear weapon race. However, these numbers do not mean that Russia's nuclear force is more modernized.
In conclusion, both countries are modernizing their nuclear power in different ways. Comparing the two countries doesn't make sense because they are both developing new modern nuclear weapons; and some military information is confidential and unobtainable currently.