The ADC of all 8 bit PIC families is a hardware module. You set it up by configuring various bits in a couple of registers and enable it with the ADON control flag. Then to perform a conversion you select an input channel to convert, wait briefly for the input to stabilise after switching input channels (or if converting a single channel, you can initialise it with that channel selected and skip further channel selection and the brief wait), then trigger the conversion by setting the Go_/Done control flag. The conversion happens autonomously, and its completion and the presence of an updated result in the ADRESH:ADRESL register pair is signalled by it clearing the Go_/Done control flag and setting the ADIF interrupt flag.
Except under certain special circumstances involving the FRC conversion clock and SLEEP mode, it doesn't matter what code the CPU core is running while the ADC is converting. This includes going off to service your Timer 1 interrupt. As an ADC conversion takes at most a few uS, the odds are the conversion will complete while the Timer 1 ISR is running, but the usual ADC code that loops polling the Go_/Done flag wont even notice - it will just see conversion complete and carry on to fetch the result.
By using a CCP module in compare: special event mode, the ADC module can run almost entirely autonomously, performing an ADC conversion at a predetermined time without further CPU core intervention. However you do then need an ADC ISR to handle fetching the result and setting the channel and time for the next conversion.