Non-deterministic TM and problems in class P
The class $P$ is a subset of $NP$. For an $NP$ algorithm to simulate $P$ in polynomial time, it does not even need to use nondeterminism, since a single branch can solve the $P$ problem. 

It could be a bit confusing though when textbooks say that an $NP$ algorithm works in 'polynomial time'. But this refers to the amount of time it takes for a single branch to compute. But taken as a whole, $NP= \cup NTIME(n^k)$, while $P= \cup DTIME(n^k)$.