No. $(2n+1,2m+1) = (17,51)$ is a counterexample because $$ 2^{51} - 1 = 7 \cdot 103 \cdot 2143 \cdot 11119 \cdot 131071 $$ and $131071 = 2^{17}-1$. The only other counterexample with $m \leq 100$ is $(2n+1,2m+1) = (37,111)$ with $P_0 = (2^{37}-1)/223 = 616318177$.

P.S. The condition $2^{2n+1} - 1 \mid 2^{2m+1} - 1$ is equivalent to $2n+1 \mid 2m+1$.

No. $(2n+1,2m+1) = (17,51)$ is a counterexample because $$ 2^{51} - 1 = 7 \cdot 103 \cdot 2143 \cdot 11119 \cdot 131071 $$ and $131071 = 2^{17}-1$ is prime. The only other counterexample with $m \leq 100$ is $(2n+1,2m+1) = (37,111)$ with $P_0 = (2^{37}-1)/223 = 616318177$.

P.S. gp code:

forstep(k=3,200,2,if(!isprime(k),f=factor(2^k-1)[,1]; r = znorder(Mod(2,f[#f])); if(r<k, print([r,k]))))

P.P.S. The condition $2^{2n+1} - 1 \mid 2^{2m+1} - 1$ is equivalent to $2n+1 \mid 2m+1$.