POVM for first m qubit measurement

Well, since these are projective measurements on the subspace of the first $m$ qubits, we can just list all projectors on the computational basis of this first subspace and 'pad' them with $I$'s on the second subspace:

$$ P_{j} = |jranglelangle j|_{m} otimes I_{|n|},,,, forall j in {0,1}^{m}, $$ which gives exactly $|{0,1}^{m}| = 2^{m}$ different operators for the POVM. If you identify distinct measurement outcomes with every operator, say $lambda_{j} = j_{d}$ (e.g. $j$ in decimal form), you can easily write down a measurement operator as well:

$$ M = sum_{j} lambda_{j}P_{j} = sum_{j} j_{d}|jrangle langle j otimes I_{n}| $$

See also, for instance, this nice answer by Daftwullie for a different measurement operator. Note that that answer omits the extra subspace of $n$, but you can just treat that by padding with $I$'s again.