Widgets of Type A arrive with Poisson Process with arrival rate $\lambda_A$, and, for Type B, with arrival rate $\lambda_B$ (independent).
During $t,$ there have been $b$ arrivals of Type B. What are the expected arrivals of Type A+B in time frame $t$?
Does one simply take the given value $b,$ and add to that the expected arrivals for process A?:
$$b+t \times \lambda_A$$
In general, $N_t=N^A_t+N^B_t$ implies $\mathbb E(N_t\mid N^B_t=b)=\mathbb E(N^A_t\mid N^B_t=b)+b$. If furthermore the processes $N^A$ and $N^B$ are independent, then $\mathbb E(N^A_t\mid N^B_t=b)=\mathbb E(N^A_t)=\lambda_At$.
Thus, in your setting, $\mathbb E(N_t\mid N^B_t=b)=b+\lambda_At$.