Trump is lying. The US is still using the Minuteman III and Trident II from the Cold War. Those missiles are ballistic so can be intercepted by the latest ABM. And adding tail kits to B61 bombs is hardly difficult.
I think both the refurbishment of Trident II and the tail kits for the B61 were started when Obama was POTUS.
Russia has more warheads. They have the Yars and Bulava in service. For tactical nukes there is the Kh-102, Zircon, and Kalibr. There is simply no contest.
It is also claimed Russia kept miniaturizing its warheads after the Cold War unlike the US. So they are more compact.
		
		
	 
Removing the nuclear ICBM counterpart, the rest of the modernization is in advanced stages.
The LRSO missile is almost ready.
The B-21s are progressing well.
The B12/13 bombs are progressing.
The B-52Js are also progressing.
The new W93 warheads are following suit.
The Tridents have been modernized.
The SLCM-N, which strictly speaking is not a component of the nuclear triad, being considered a tactical weapon, is also advancing.
The problem continues in the Sentinel ICBM program, but it is progressing.
In the next decade, the American triad will consist of:
12 Columbia SLBMs;
240 Trident D5LE missiles;
Approximately 1200 W76-1, W76-2, W88, and W93 warheads;
400 Sentinel ICBMs;
Approximately 400/600 W87-0/W87-1 warheads;
75 B-52J bombers;
100 B-21 bombers;
1000 ALCM LRSO missiles with W80-4 warheads;
Approximately 1000 bombs (B61-13, B61-7, B61-11, B83);
– Tactical arsenal:
Approximately 60 SSNs;
Approximately 300 SLCM-N missiles with W80 warheads;
Approximately 1,800 F-35A fighters;
Approximately 500 B61-12 bombs;
Some F-15E and F-16 fighters can also be nuclear weapons vectors, including "strategic" bombs.