You cannot use Astral Spell from an area that blocks planar travel
Private Sanctum states as one of the options which I assume you chose (otherwise there is no question about it, nothing would be blocked):
Planar travel is blocked within the warded area.
The spell proofs the area against planar travel, and therefore prevents creatures from accessing the area by way of the Astral Plane, (...)
And finally Astral Projection states:
You and up to eight willing creatures within range project your astral bodies into the Astral Plane (...) Your astral form can freely travel through the Astral Plane (...) If you enter a new plane or return to the plane you were on when casting this spell, your body and possessions are transported along the silver cord, allowing you to re-enter your body as you enter the new plane. (...) When the spell ends, the affected creature returns to its physical body, and it awakens.
You cannot leave
You will not be able to project your astral body out of the warded area, as this would be planar travel.
You cannot reunite with your body
What would happen if your unconscious body was carried into an area protected by these wards after you cast the spell? Could you return? I.e. does the unwinding along your silver cord also count as astral travel?
Forbiddance explicitly refers to blocking creatures from accessing the area by way of the Astral Plane, and your astral body is referred to as a creature. Thus, your astral body would not be able to enter the area.
The reason given for this in Forbiddance is that the area is protected from planar travel, just as Private Sanctum's area. Therefore your astral body would also not be able to re-enter the Private Sanctum.
Your DM will need to rule what happens
Astral Projection says about the silver cord
If the cord is cut--something that can happen only when an effect specifically states that it does -- your soul and body are separated, killing you instantly.
Neither of the other spells specifically states this, so the cord would remain unbroken.
The spell does not foresee what happens if you enter a plane, and you cannot be re-united with your body. This would be the domain of rulings from your DM.
They could for example rule that the silver cord constitutes an exemption to the general rule, as normally you or other creatures are not connected by a silver cord to the plane they enter via astral projection. Or they could rule that your projection is stuck outside of the warded area, unable to move and unable to reunite with your body. Or something else. The rules do not resolve this edge case.
The astral plane is a bad place for retirement
An additional comment because your stated motivation for going to the astral plane is immortality for your high-level cleric: the Dungeon Masters Guide on page 47 describes the astral plane like this:
Only a few things can sever a traveler's silver cord, the most common being a psychic wind. (...) A psychic wind is made up of lost memories, forgotten ideas, minor musings, and subconscious fears that went astray in the Astral Plane and conglomerated into this powerful force.
If you stay too long on the astral plane, you run the risk that one of these winds disrupts your silver cord and instantly kills you. Since your astral cord trails all the way through the plane to connect you to your body, Temple of the Gods will not be able to protect it.
Furthermore on page 48:
The most prominent denizens of the Astral Plane are the githyanki, an outcast race of reavers that sail sleek astral ships, slaughter astral travelers, and raid planes touched by the Astral.
These guys have silver swords that can cut your astral cord and instantly kill you. As a level 17 cleric you may be able to stand up to them, and at least they have no direct way into your Temple, but their Gish wizards might Dimension Door them in.
Characters who linger for too long in the Astral might have an encounter with one or more wandering angels, demons, devils, night hags, yugoloths, or other planar travelers.
This does not sound like a golden retirement to me, and clearly points out the risks of prolongued tenancy. Don't complain if your DM decides these risks are very real. The kind of creatures that roam the astral plane such as a lich may well be able to disintegrate your Temple.
As a last note, on the home front, nothing in Private Sanctum stops normal creatures from entering its area. You would need to combine it with other protective measures to secure the resting body.