India mastered nuclear warhead integration on missiles in the 1990s, according to revelations from the book " Weapons of Peace" by Raj Chengappa, who interviewed Abdul Kalam, Homi Sethna, Raja Ramanna, R Chidambaram and many other key people of the Indian nuclear programme. We first worked towards making nuclear weapons deliverable by air by mating a bomb to a Mirage 2000, which was successfully validated in 1994. Then we made the warhead of Prithvi nuclear capable in 1997, and then we tested an Agni 2 with a full nuclear configuration (without the plutonium core) successfully in 1999. Since our Agni 3, 4, and 5 can carry much higher payloads than Agni 2, we can easily say we have successfully mated our nuclear weapons on all of our strategic missiles.
As for the range, it depends on the warhead type and yield. These are top secret details, we do know India has 100% attained mastery and weaponisation of the atomic bomb and likely the boosted fission bomb. And although there are doubts, I firmly believe we have attained mastery over hydrogen bombs of up to 200 kilotons. The weight of these bombs is unknown, but for thr hydrogen bomb, it's likely a lot less than the 2 tonne maximum that the Agni 5 can carry. Thus, Agni 5 can deliver any atomic weapon we have over it's full range and likely more even.
I highly recommend you to read "Weapons of Peace" by Raj Chengappa. It is an extremely detailed book on the Indian nuclear bomb and missile programme. It revealed that we tried to make a 8,000 km ICBM way back in the 1970s!!! You can read thr book here:
Weapons of peace : the secret story of India's quest to be a nuclear power : Chengappa, Raj : Free Download, Borrow, and Streaming : Internet Archive
I have also attached some screenshots of specific pages in the book which details what I have said.