Introduction:
The Lodi dynasty in India ruled from 1451 to 1526, and their reign ended with the invasion of Babur, the founder of the Mughal Empire. During this time, several Bhakti saints emerged and spread their teachings across the country. Among these saints were Dadu Dayal, Guru Nanak, and Tyagaraja. This question asks which of these saints were preaching when the Lodi dynasty fell and Babur took over.

Explanation:
To determine the correct answer, let's look at the timeline of these Bhakti saints and the historical events surrounding the fall of the Lodi dynasty and the rise of Babur.

Dadu Dayal:
Dadu Dayal was a prominent saint and poet who lived during the 16th century. He preached in the region of Rajasthan and Gujarat. However, he was born in 1544, several years after the fall of the Lodi dynasty. Therefore, Dadu Dayal was not preaching during the time of the Lodi dynasty's downfall and Babur's rise.

Guru Nanak:
Guru Nanak was the founder of Sikhism and lived from 1469 to 1539. He was born in Punjab, which was the heartland of the Lodi dynasty. Guru Nanak began his spiritual journey and preaching in the late 15th century. His teachings emphasized equality, social justice, and devotion to one God. The Lodi dynasty fell in 1526, and Guru Nanak was actively preaching during this period. Therefore, Guru Nanak was preaching when the Lodi dynasty fell and Babur took over.

Tyagaraja:
Tyagaraja was a renowned composer and one of the greatest exponents of Carnatic music. He lived from 1767 to 1847, which was long after the Lodi dynasty had fallen and the Mughal Empire had been established. Therefore, Tyagaraja was not preaching during the time of the Lodi dynasty's downfall and Babur's rise.

Conclusion:
Based on the above analysis, we can conclude that Guru Nanak was the only Bhakti saint who was preaching when the Lodi dynasty fell and Babur took over. Therefore, the correct answer is option 'B' - Guru Nanak only.