Soundness of the fast of one who is junub

Firstly: 

What you should have done, if you were unable to do ghusl,
was to do tayammum (“dry ablution”), because tayammum is prescribed for
major impurity, i.e., janaabah, menses and nifaas (postpartum bleeding)
according to the majority of scholars; it is also prescribed in the case of
minor impurity, according to scholarly consensus. 

See: al-Majmoo‘ Sharh al-Muhadhdhab by an-Nawawi,,
2/207 

The evidence for that is the hadeeth about the man who did
not pray with the Prophet (blessings and peace of Allah be upon him). When
he had finished praying, he said to him: “O So and so, what prevented you
from praying with the people?” He said: I became junub and had no water. He
said: “You should use clean earth, for it will suffice you.” Narrated by
al-Bukhaari (344) and Muslim (682). 

See also the answers to questions no.
40204 and
87711

Secondly: 

Purification is not a condition of fasting being valid.
‘Aa’ishah and Umm Salamah (may Allah be pleased with them) narrated that
dawn would come and the Prophet (blessings and peace of Allah be upon him)
would be in a state of janaabah, having been intimate with his wife, then he
would do ghusl and fast. Narrated by al-Bukhaari (1926) and Muslim (1109).
According to Muslim, it says: “not as the result of an erotic dream”. If
that is the case with regard to intercourse, then it applies also to a wet
dream and is more appropriate, because intercourse happens by the
individual’s choice, whereas he has no choice with regard to a wet dream
(i.e., it occurs involuntarily). 

The scholars unanimously agreed that the fast of one who is
junub because of a wet dream is valid. Al-Maawardi said: The ummah is
unanimously agreed that if a person has a wet dream at night and he is able
to do ghusl before dawn, but he does not do ghusl and he starts the morning
junub as a result of a wet dream, or he has a wet dream during the day, his
fast is still valid. End quote from al-Majmoo‘ (6/308). 

Thirdly: 

You made a very serious mistake, which is that you prayed
without having purified yourself. Purification from impurity is one of the
conditions of prayer being valid according to scholarly consensus, as was
narrated by Ibn al-Mundhir in al-Ijmaa‘ (1) and an-Nawawi in Sharh Muslim
(3/102). 

What you should have done was ask knowledgeable people so
that you would not fall into such an error. The ways of asking questions
have become easier nowadays, praise be to Allah, and finding out Islamic
rulings has become a simple matter that is not difficult for anyone. If you
fell short in finding out about the ruling on the matter, then you have to
repent to Allah and ask for forgiveness for your sin. If you did not fall
short, then Allah will excuse you for your ignorance, by His grace and
kindness. 

But there was a difference of opinion among the scholars as
to whether you have to make up those prayers that you offered without
purification, or not. 

The most correct view is that this comes under the heading of
cases in which one may be excused for ignorance or mistakes, because the
person in question was confused on the basis that he was not able to do
gusl, so he thought that he did not have to do ghusl, and he did not know
that tayammum is an alternative to ghusl in the case of janaabah. 

This view was supported by Shaykh al-Islam Ibn Taymiyah, who
said in Majmoo‘ al-Fataawa (23/37): With regard to the one who did
not know that (some action having to do with the prayer) was obligatory,
then as soon as he finds out he should offer the prayer that is due at that
time and subsequent prayers (with that obligatory action), but he does not
have to repeat any of his previous prayers. It is proven in as-Saheehayn
that the Prophet (blessings and peace of Allah be upon him) said to the
Bedouin who prayed badly: “Go back and pray, for you have not prayed.” He
said: By the One Who sent you with the truth, I cannot do any better than
this. Teach me that which will suffice me in my prayer. So he (blessings and
peace of Allah be upon him) taught him, and he instructed him to repeat the
prayer that was due at that time, but he did not instruct him to repeat
prayers that he had offered in the past, even though the man said: I cannot
do better than this. By the same token, he did not instruct ‘Umar and
‘Ammaar to repeat the prayer; when ‘Umar became junub he did not pray, and
‘Ammaar rolled in the dust like animals do. And he did not instruct Abu
Dharr to do the prayers that he has missed when he was in a state of
janaabah. And he did not instruct the woman who was suffering from irregular
non-menstrual bleeding (istihaadah) to make up the prayers that she had not
done, even though she said: I bleed very heavily and it prevents me from
fasting and praying. And he did not instruct those who ate during Ramadan
until the white string became distinct from the black string to repeat the
fast. When the prayer was first enjoined, it was two (rak‘ahs) by two, then
when Zayd migrated, the prayer when not travelling was made four rak‘ahs. In
Makkah, Abyssinia and the deserts there were many Muslims who did not know
about that until after some time had passed, and they continued to pray two
rak‘ahs, but he did not instruct them to repeat the prayers that they had
offered. Similarly, he did not instruct those who had been praying towards
the abrogated qiblah (Jerusalem) to repeat the prayers they had offered
facing that direction before news of the abrogation had reached them. 

Then he said: A number of the senior Sahaabah used not to do
ghusl after being intimate with their wives if no ejaculation occurred,
because they thought that water (ghusl) was for water (emission of semen),
until it was proven to them that that ruling had been abrogated. For some of
them, it was not proven to them that it was abrogated, and they continued to
pray without the purification that is required according to Islam, because
they did not know that it was obligatory, and one of them would pray when he
was in a state of janaabah. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked
about a man who did wudoo’ and prayed Fajr when he was in a state of
janaabah, because he was afraid that he would miss the prayer in
congregation. He replied: 

The correct view is that if he did that out of ignorance,
then he is to be excused for that, as the Prophet (blessings and peace of
Allah be upon him) excused the one who prayed badly and did not instruct him
to make up past prayers; and he excused the woman who was suffering from
irregular non-menstrual bleeding (istihaadah) who did not pray; and
he excused ‘Ammaar ibn Yaasir when he rolled in the dust, thinking that this
was what was required of him for tayammum. And there are many similar
reports.

End quote from Liqa’ al-Baab al-Maftooh, no. 54 

See also the answers to questions no.
21806,
45648,
150069

And Allah knows best.