How long will it take for hydroquinone 10% to work on stubborn melasma?

No two person's physiology are exactly the same.  Even melanin - the pigment in our skin - can vary from person to person.  For sure, every person's physiological response to a medication varies.Thus, I can't tell you precisely how your skin will respond to Hydroquinone 10%.   However, I can tell you how it is supposed to work!
Hydroquinone is applied to the surface of the skin.  From there it leaches down to the bottom of the epidermis where the melanocyte cells live, on the border between the epidermis and the dermis. Hydroquinone does its work by suppressing the melanocytes ability to make melanin.  Thus new epidermis cells (epithelial cells) will contain less melanin in them when they reach the surface and can be seen.  As it takes about 3 weeks for these new epithelial cells to reach to the surface, you should start seeing the effects of the Hydroquinone in approximately 3 weeks.
If the Hydroquinone is not effective for you, know that our customized laser protocols have generated excellent results on even the toughest melasma cases for nearly 20 years.

HQ ideally should not be used as monotherapy, however in the context of your question, it should work within 6-8 weeks.Melasma can not be completely cured, however effective treatment are possible. Melasma patients have skin that is very sensitive to UV light. Even the smallest amount can stimulate your pigment cells to produce colour. The mainstay of treatment is strict UV protection- hats, sunglasses and SPF every 4 hours. 
I prefer to use a combination of creams and laser for melasma. My most frequently prescribed topicals include hydroquinone, along with vitamin A creams and botanical extracts. The exact percentages will depend on your skin sensitivity, clinical presentation and your current skin care routine. 
I usually start my patients on a series of lasers (low dose Q switch or Picosure), along with medical therapy. Another laser I use is the Clear and Brilliant in the Permea setting. In some cases I combine glycolic AHAs, TCA low strength single coat peels, and in other cases I use a tablet called Tranexamic acid to help modulate the blood vessels that are associated with melanocytes (pigment cells). Melasma treatments should be tailored to each patient. 
All the best, Dr Davin Lim. Laser, surgical and aesthetic dermatologist. Brisbane, Australia. Cutis International