China has had nuclear subs for decades. It is just that their subs were attack subs rather than strategic nuke subs. From what I have heard, they already own a few nuke subs, and they are simply working on install the necessary systems for load nuke missiles onto them.
Even if all of it is successful, they still face multiple challenges.
1. Their subs are said to be quite noisy, thus making them easier to be detected.
2. Their numbers are limited so far. Until they mass produce at least a dozen or so of these ballistic missile subs, they will not have meaningful deter against other nuclear powers.
3. They are surrounded by the so called "first island chain" and "second island chain". Basically the US military has bases all over the places. If Chinese subs need to leave their home turf, they first need to go through a few (limited number of) bottle necks.
4. They still need to solve the problem of long range communication with their hiding subs far away from home.