He married a woman before the end of her ‘iddah following the death of her husband

Firstly: 

What is
required of a woman after her husband dies is to observe an ‘iddah of four
months and ten days from the date of his death, if she is not pregnant,
because Allah, may He be exalted, says (interpretation of the meaning):

“And
those of you who die and leave wives behind them, they (the wives) shall
wait (as regards their marriage) for four months and ten days”

[al-Baqarah
2:234]

If she is
pregnant, then her ‘iddah lasts until she gives birth, because Allah may He
be exalted, says (interpretation of the meaning):

“And for
those who are pregnant (whether they are divorced or their husbands are
dead), their Iddah (prescribed period) is until they deliver (their
burdens),”

[at-Talaaq
65:4]. 

If she gets
married during the ‘iddah, then her marriage is invalid according to
scholarly consensus, and they must be separated. 

Ibn Qudaamah
(may Allah have mercy on him) said:

It is not
permissible for the woman who is in ‘iddah to get married during her ‘iddah,
according to scholarly consensus, no matter what kind of ‘iddah it is,
because Allah, may He be exalted, says (interpretation of the meaning):

“And you
should not settle anything finally about the marriage until the waiting term
expires”

[al-Baqarah
2:235].

If she does
get married, then the marriage is invalid. 

End quote
from al-Mughni, 8/100 

It is
obligatory to separate her and her second husband, then for her to complete
her ‘iddah from scratch. Then when she has completed the ‘iddah for the
first marriage, it is permissible for the second husband to make a new
marriage contract with her. 

Ibn Qudaamah
(may Allah have mercy on him) said:

Once this is
established, then he had to leave her and if he does not do that, then they
must be separated. If he leaves her or they are separated, then she has to
complete the ‘iddah for the first marriage, because that duty takes
precedence and that ‘iddah is obligatory because the marriage to the first
husband is proven and is valid. Once the ‘iddah for the first marriage is
completed, then she has to observe an ‘iddah for the second marriage, and
the two ‘iddahs cannot be combined because they are for marriages to two
different men. This is the view of ash-Shaafa ‘i. … Then he stated that this
opinion was narrated from ‘Umar ibn al-Khattaab and ‘Ali ibn Abi Taalib (may
Allah be pleased with them both), then he said: These two opinions are known
from two prominent caliphs and it is not known that any of the Sahaabah
disagreed with them. Moreover they are rights that belong to two different
people, and they cannot be combined.

End quote
from al-Mughni, 8/101 

Then after
the ‘iddah for the first marriage is complete, the second husband may do a
marriage contract with her as soon as the ‘iddah for the first marriage has
ended. 

Ibn Qudaamah
(may Allah have mercy on him) said: ash-Shaafa ‘i said in al-Jadeed:
He may marry her after completion of the ‘iddah following the first marriage
and there is no reason why he should not marry her during her ‘iddah from
him (the second husband) because the ‘iddah was only prescribed so as to
protect lineages, and the lineage belongs to him in this case. This is good
and it makes sense.

End quote
from al-Mughni, 8/100 

Shaykh Ibn
‘Uthaymeen (may Allah have mercy on him) said: 

In
principle, the more correct view is that she becomes permissible for him
with a marriage contract after the end of the first ‘iddah, especially if he
has repented to Allah, may He be exalted, because the ‘iddah is for him
(i.e., following the end of a marriage to him).

End quote
from ash-Sharh al-Mumti‘, 13/383 

Secondly: 

Even though
this marriage is invalid, his children are to be attributed to him because
it was like a marriage, and that was due to being unaware of the Islamic
ruling. 

Ibn Qudaamah
(may Allah have mercy on him) said: 

If a women
who is observing ‘iddah gets married and they were aware of the ‘iddah and
the prohibition on marrying during this period, and he had intercourse with
her, then they are zaanis; the hadd punishment for zina should be carried
out on them, she is not entitled to any mahr and any child born should not
be attributed to him. But if they were unaware of the ‘iddah or of the
prohibition, then the child is to be attributed to him, the hadd punishment
is waived and the mahr must be paid.

End quote
from al-Mughni, 8/103 

And Allah knows best.