In general, under English law, property, titles etc will be inherited by a child who is in utero at the time of the father’s death, provided he is subsequently born alive. So if this rule does not apply to the crown, it is an exception to the general rule.
Having said that, you can see a case for such an exception. There is also a rule that the crown can never be vacant. My guess would be that, if the situation ever arose, the courts would rule that the crown would pass to the heir presumptive, and that a child not born could not inherit the crown.
Members of the Royal family could not take seats in the House of Lords unless they had peerages of their own, which most do not. Those who do have peerages (e.g. the Duke of York) have hereditary peerages rather than life peerages, and a hereditary peerage no longer guarantees a seat in the House of Lords, so they would need to be elected by the other hereditary peers.
Legally, so far as I know, there is nothing to stop a member of the Royal family who has no peerage (e.g. Princess Anne) from seeking election to the House of Commons. And, legally, the Prime Minister can be drawn either from the Lords or the Commons – in fact he doesn’t have to be a member of either house - so a Royal, whether or not a member of either House, could be appointed Prime Minister
But this would all be regarded as constitutionally highly undesirable, and none of it is likely to happen.