To simplify, denote $y(t)=\log N_t$. Then we have a simple differential-difference equation
$$y'(t)=f(y(t-\tau)),\quad f(y)=r(1-y/K).$$
Let $\phi$ be an arbitrary continuous function on $[0,\tau]$
which has one sided derivative at $\tau$ satisfying $\phi'(\tau-)=\phi(0)$.
Then the formulas $Y(t)=\phi(t), 0\leq t\leq \tau$, and
$$y(t)=y(n\tau)+\int_{n\tau}^t f(y(t-\tau))dt, \quad n\tau\leq t\leq (n+1)\tau,\quad n\geq 1$$
defines a continuous solution which is everywhere continuous and continuously differentiable for $t\geq \tau$. Then $N_t=\exp(y(t))$ defines a solution of your problem.