The residual radioactivity decays at the same rate, but there's also more of it (roughly proportional to weapon yield). Since modern weapons yield much more energy than the bombs of 1945 they also yield much more radioactivity. Further, surface or ground-penetrating attacks against military targets would tend to rain out much more of the radioactive material locally than the relatively high altitude airburst strikes of 1945. The Japanese targets would have probably suffered fewer casualties but would have had a more persistent radioactive contamination problem if the 1945 bombs had been configured to explode on hitting the ground instead of in the air.
You might expect that modern weapons would have lower yields of radioactive fission products since they all incorporate fusion and don't rely on fission alone. And it's true that the US and USSR (and perhaps other nations) successfully developed thermonuclear weapons that produced over 95% of their energy from fusion, which doesn't create radioactive fission products. This was in the late 1950s/early 1960s.
This is an amazing recently published article about the cleanest American device ever tested:
"Ripple: An Investigation of the World’s Most Advanced
High-Yield Thermonuclear Weapon Design"
However, devices like that described above were too large for submarine launched missiles. The most effective way to make small, light, efficient warheads for submarine launched missiles is to use the fusion reaction's fast neutrons to fission additional uranium in the tamper around the fusion fuel. So modern submarine launched thermonuclear weapons develop about half of their explosive yield from fission, with corresponding generation of radioactive fission products.
