Tried my best to make a list of rulers for Madras. I'm no expert, just interested, but hope it's a help :D
It seem like a lot of coins were minted posthumously for Alamgir II.
The line between the East India Company and the British Empire is quite blurry at times.
Mughal Empire: Madras Presidency
Timurid dynasty - House of Babur
- Aurangzeb/Alamgir (1658-1707)
Wiki: Q485547
Bahadur Shah I/Shah Alam I (1707-1712)
Wiki: Q544228
- Farrukh Siyar (1713-1719)
Wiki: Q559490
Example: N#75249
- Muhammad Shah (1719-1748)
Wiki:Q558714
Eksample: N#66319
- Ahmad Shah Bahadur (1748-1754)
Wiki: Q400807
Example: N#58340
- Alamgir II (1754-1759)
Wiki: Q732026
Example: N#42731
- Shah Jahan III (1759-1760)
Wiki: Q2627138
Example: N#82530
- Shah Alam II (1760-1788 & 1788-1806)
Wiki: Q719449
Example: N#82530
Dynasty: Timurid dynasty - House of Babur - Puppet of the East India Company
- Akbar Shah II (1806-1837)
Wiki: Q2091579
Example: N#25912
Dynasty: Timurid dynasty - House of Babur
- Bahadur Shah II Zafar (1837-1857)
Wiki: Q127369
Example:
East India Company (1797..???)
Wiki: Q83164
Example: N#50417
British Empire - House of Hannover
- George III (1760-1820)
Wiki:Q127318
Example: N#47580
- George IV (1820-1830)
Wiki: Q130586
Example: N#329508
- William IV (1830-1837)
Wiki: Q130822
Example: N#35941
- Victoria (1837-1901)
Wiki: Q9439
Example: N#219985
Referee who may have an opinion: @SStefanov
Sources:
https://en.wikipedia.org/wiki/List_of_emperors_of_the_Mughal_Empire