If your spouse is here as your dependant he or she can study either full-time or part-time and do not need to have immigration status as a student.They will be subject to the same rules for fees assessment and student support as you. See the UKCOSA website for more information.
Children who are in the UK as the dependants of an international student are also allowed to study full-time or part-time. Those aged 16 and under can go to state schools without paying any fees and may be able to remain at school until 18 if they started at that school at the age of 16 or under.