When he was praying Zuhr, he remembered that he had not prayed Fajr

This question will be answered by addressing three points: 

1 – The ruling on praying missed prayers in order: 

Imaam Abu Haneefah, Imaam Maalik and Imaam Ahmad are of the
view that it is obligatory to pray missed prayers in order when making them
up. The evidence for that is the fact that when the Prophet (peace and
blessings of Allaah be upon him) missed some prayers on the day of
al-Khandaq, he made them up in order. Al-Bukhari (641) and Muslim (631)
narrated from Jaabir ibn ‘Abd-Allaah that the Prophet (peace and
blessings of Allaah be upon him) prayed ‘Asr on the day of al-Khandaq after
the sun had set, then he prayed Maghrib after that. Al-Bukhaari (631)
narrated that the Prophet (peace and blessings of Allaah be upon him)
said: “Pray as you have seen me praying.” (al-Mughni, 2/336) 

2 – If a person forgets
to pray them in order, does that obligation no longer apply? 

The answer is, yes, the obligation to pray them in order no
longer applies if one forgets, because the Prophet (peace and blessings
of Allaah be upon him) said: “Allaah will forgive my ummah for their
mistakes and what they forget, and what they are forced to do.” (Narrated by
Ibn Maajah, 2043; classed as saheeh by al-Albaani in Saheeh Ibn Maajah,
1662). This is the view of Imaam Abu Haneefah and Imaam Ahmad (may Allaah
have mercy on them). See Fath al-Qadeer, 1/424; al-Mughni, 2/340;
al-Sharh al-Mumti’, 2/139. 

If a person forgets one
prayer and does not remember until the time for the next prayer begins, then
he remembers it, one of the following three scenarios must apply: 

1-    
He remembers the missed prayer
before he starts to offer the current prayer. In this case he has to start
with the missed prayer, then pray the prayer that is currently due.

2-    
He prays the current prayer and
completes it, then he remembers that he still has to do the missed prayer
that he has not prayed yet. His current prayer is valid and he only has to
pray the missed prayer, and he is excused for not praying them in the proper
order because he forgot.

3-    
He remembers during the current
prayer that he did not pray the previous prayer, which he missed. So he
should complete the current prayer, which then becomes a naafil
(supererogatory) prayer, then he should pray the prayer his missed, and then
repeat the current prayer, in order to keep them in the proper order. This
is the view of Imam Ahmad (may Allaah have mercy on him). See al-Mughni,
2/336-340). This is also the view of ‘Abd-Allaah ibn ‘Umar (may Allaah be
pleased with them both). Imaam Maalik narrated in al-Muwatta’ (408)
that Naafi’ ibn ‘Abd-Allaah ibn ‘Umar used to say, “Whoever forgets a prayer
then remembers it only when he is with the imaam, then when the imaam says
the salaam, let him pray the prayer that he had forgotten, then let him pray
the other after that.” Shaykh al-Islam (Ibn Taymiyah – may Allaah have mercy
on him) said: “When he remembers the prayer that he missed during the
(current) prayer, it is as if he remembered before he started it. But if he
does not remember until the current prayer is finished, then the current
prayer is valid according to the majority of scholars, such as Abu Haneefah,
al-Shaafa’i and Ahmad…” (al-Fataawa al-Kubra, 1/112) 

His completing the prayer that he is currently praying is
mustahabb (recommended), not waajib (obligatory). If he stops that prayer,
then prays the prayer he missed, then prays the current prayer, that is
permissible. Muhanna said: “I said to Ahmad, ‘I was praying ‘Isha, then I
remembered that I had not prayed Maghrib. So I prayed ‘Isha’, then I prayed
Maghrib and repeated ‘Isha’.’ Ahmad said, ‘You did right.’ I said, ‘Should I
not have stopped praying when I remembered?’ He said, ‘Yes.’ I said, ‘So how
did I get it right?’ He said, ‘Both are permissible.’” (al-Mughni,
2/339). 

Some of the scholars are of the view that he should complete
the (current) prayer that he is praying, and then do the missed prayer, and
he does not have to repeat the current prayer. This is the view of
al-Shaafa’i, as stated in al-Majmoo’, 3/70. This was also the view
favoured by Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him). (Majmoo’
Fataawa Ibn ‘Uthaymeen, 12/221). But the first opinion is more on the
safe side. 

3 – The questioner says that he did not pray Fajr because
he was sick 

What he did was not correct, because sickness is not an
excuse to delay the prayer until the time for prayer is over. Rather what
the Muslim must do is to offer the prayer on time. If he is sick, he should
pray as best he can, for Allaah does not burden a soul beyond what it is
able to bear. If he is unable to stand, then he may pray sitting down. If he
is unable to sit, then he may pray lying on his side. If he is unable to do
wudoo’, then he should do tayammum. If there is some impurity on his body
and he is unable to remove it, then he should pray as he is, and so on. It
is not permissible for him to delay the prayer until the time for it is over
because he is unable to purify himself or remove the impurity. Rather he
should pray as best he can and do as much of the obligatory actions of
prayer as he is able, and he is relieved of the obligation to do the things
that he is unable to do. See the essay Ahkaam Salaat al-Mareed wa
Tahaaratuhu by Shaykh Ibn Baaz.