What is required for an English child to be legitimate for purposes of
inheriting titles or the Crown?

The canon law recognizes legitimation by subsequent marriage. But some
time, I think in 13th century, the Church asked the English to accept
the canon law regarding legitimation of bastards. The peers of England
refused to change the laws of England as the Church requested. Thus,
bastards cannot inherit English titles, or the throne, even if their
parents then marry.

Scottish law does recognize legitimation by subsequent marriage.

The Westminster parliament has changed English laws to allow for
legitimation of bastards - but such legitimation expressly does not
affect titles.

What about offspring of void marriages?

Canon law says that offspring of void marriages is not authomatically
bastards: if either or both of the parties were ignorant and innocent
party about the invalidity, the children are legitimate. What is the
position of the English common law?

Edward IV naturally acted in bad faith towards Elizabeth Woodville -
but did she know he was perpetrating bigamy?

Henry VIII insisted that both his daughters were bastards as inevitable
consequence of the annulments of his marriages. Even though he made a
will as allowed by Act of Parliament giving his bastard daughters
remainder to the throne, they emphatically stayed bastards. Any future
daughter of Henry by Katherine Parr or any later lawful wife of his
would have inherited the Throne ahead of her elder but bastard sisters.

Also, there are pretty weird and confusing decisions of Lords about
what constitutes paternity. One peeress of early 17th century left her
husband and openly lived with a lover; when her son naturally assumed
that he had nothing to do with her husband and therefore was a
commoner, the Lords declared that it did not matter and that he as the
legal son of her husband was his heir and a peer whether he wished or
not. Another peeress, the wife of someone surnamed Knollys, stayed by
her husband until he died and bore one son in his lifetime, another
soon after his death. It took the Lords nearly two centuries to decide
that the boys were not her husband´ s!

So...

Does it mean that children of a bigamous marriage are bastards and
cannot inherit titles or Throne irrespective of what the parties to the
marriage knew?

At least, people subject to RMA cannot contract secret marriage. They
can marry either by royal consent recorded by Privy Council or else
record their wish to marry with Privy Council and marry at least a year
later. Any other marriages are null and void. The place of celebration
is irrelevant, because RMA follows people who are subject to it
wherever they might go. A secret marriage by someone subject to RMA
needs no formal divorce or annulment and is no bar to a subsequent
marriage under RMA. The marriage of George IV and Queen Caroline was
perfectly valid and not bigamous. The punishments of praemunire for
assisting celebrations of marriages against RMA have been repealed, so
marriages against RMA are innocent fun of no legal importance.

But RMA does not apply to nonroyals. Which means that a marriage duly
celebrated under RMA might still be bigamous, null and void if, unknown
to the Privy Council, the nonroyal party has contracted secret but
legally valid marriage before.

How would evidence that some royal marriage was bigamous due to a
secret marriage by the nonroyal party affect the styles and titles of
the offspring?

Also... The paternity of Knollys was challenged and denied nearly two
centuries after his death.

What, constitutionally, would happen if someone were to tender evidence
questioning whether the then Duke of Kent was the father of Victoria?

Also, what is the position of English law regarding the impediment of
consanguinity?

There has been some rumours that Diana was not the daughter of Earl
Spencer.

Denying that she was his daughter would not affect very much, legally.

But what would happen to the legitimacy of the princes of Wales if
someone were to accuse prince Philip of being the father of Diana?