Problem in deriving Noise Figure for Cascaded systems
When you think of the noise figure of each individual block, you assume that the input noise is purely the thermal noise floor. The level of the input noise is important in the definition of the NF, otherwise you would get different numbers for NF depending on the value of the input noise, since the noise internally produced (Np) is constant.

When you start with the formula for F2, you're making:

$$
F_2=\frac{N_3}{G_2N_2}
$$

This is not correct because the noise level at the input of stage 2 is much higher than the thermal noise floor (due to the amplification G1 and the additional noise created by stage 1). For a given F2, this makes N3 much higher than it actually is, what makes the cascade NF much higher than what it's supposed to be.

You should instead use:

$$
F_2=1+\frac{N_{p2}}{G_2N_1}
$$

The important part is to understand why N1 is being used here. It's because, by the definition of each individual noise figure, we have to assume that the input noise level is the thermal noise level.

Rearranging this, we get:

$$
\frac{F_2-1}{G_1}=\frac{N_{p2}}{G_1G_2N_1}
$$

Which can be replaced in your formula:

$$
F = 1 + \frac{N_{p1}}{G_{1}N_{1}} + \frac{N_{p2}}{N_{1}G_{1}G_{2}}
$$

To get the expected result:

$$
F = F_{1} + \frac{F_{2}-1}{G_{1}}
$$