TMOD value for timer 1 in mode 0 in 8051 microcontroller
Actually, all of those statements modify BOTH timers:

MOV TMOD,#00H ; set timer 0 mode 0 AND timer1 mode 0
MOV TMOD,#01H ; set timer 0 mode 1 AND timer1 mode 0
MOV TMOD,#10H ; set timer 0 mode 0 AND timer1 mode 1


If you want to modify only one of the timers, you must do a read-modify-write, preserving the bits on the timer you're not changing.

For example, if you want to force Timer0 to mode 0 without changing anything else, you might do:

ANL TMOD,#FCh ; binary 11111100 -- bit mask with two LSBs cleared