The most prominent and reputed museum in Mumbai is the Chhatrapati Shivaji Maharaj Vastu Sangrahalaya (it was formerly called the Prince of Wales museum). It is not a museum which is showcasing only the history of various eras, but also a lot of other aspects, including textiles, paintings, coins, antiques and art. You could contact the museum officials via email on email@example.com if you wish to meet one-on-one or have a guided tour for your research. The museum address is: 159-161 Mahatma Gandhi Road, Fort, Mumbai - 400023. Ph: +91 022 22844484. Website: https://www.csmvs.in/.
Visit, too, the other famous & oldest museum, Dr. Bhau Daji Lad Museum. The interior is really breath-taking and beautiful. Address: Veer Mata Jijabai Bhosale Udyan (Rani Baug), 91/A, Dr Babasaheb Ambedkar Road, Byculla East Mumbai-400027. Ph: +91 22 23731234. Website: http://bdlmuseum.org/. This museum has separate galleries for art, paintings and industrial art. You can avail of the weekend tour with the curatorial team, scheduled on Saturday and Sunday (at 11a.m. is the tour in English). There is no extra fee to avail the team's service, but you will need to pay the museum entry ticket fee.
If you are looking for a display of artwork by painters you can head to the Jehangir Art Gallery located at M.G.Road, Kala Ghoda. They hold exhibitions of prominent artists. Additionally, from time to time workshops are conducted and discussions held on various forms of art. Ph: 022- 22843989. Email: firstname.lastname@example.org. Website: https://jehangirartgallery.com/
As for a wax museum, there is just one, located in a shopping mall. This is the Red Carpet Museum. Address: R City Mall, 1st floor, 400086, Lal Bahadur Shastri Marg, Amrut Nagar, Ghatkopar West, Mumbai-400086. Ph: 022 2517 5557. Email: email@example.com. Website: http://www.redcarpetmuseum.com/
Note: It is advisable to check out the websites of the museums to know not just the timings and entry fee, but also check for announcements of days when the museum may be closed. Sometimes, a museum may be closed for maintenance or to prepare for a new exhibition or for a national holiday.
As a research scholar, you will be welcome, I am sure, so you could contact via email, if you wish, before visiting, to get some extra details for your research work. Check for internship offers also.
When people come at you with their worst, you should come at them with your best (advice given to Selena Gomez by her mother, quoted in Time magazine.)