I am planning to apply for long term tourist visa for my mom and I wanted to know how long can she stay in UK if granted a 2 or 5 year visa. And if it's 6 months, then can she come back by making an international visit for few weeks?
The holder of a long-term UK visitor visa can stay for up to 6 months on each visit. Having a long-term visa doesn’t remove the need to demonstrate that the holder is a genuine visitor as stated in the Immigration Rules V 4.2 https://www.gov.uk/guidance/immigration-rules/immigration-rules-appendix-v-visitor-rules (a) will leave the UK at the end of their visit; and (b) will not live in the UK for extended periods through frequent or successive visits, or make the UK their main home
Making visa runs will quickly lead to your mother being refused entry if Border officials think she is trying to use a visitor visa to effectively live in the UK. There is no stated period in the Rules, but as a general guide your mother should spend at least the same amount of time outside the UK as in it.