Depending on where you live depends whether switching to bt would even make a difference, unless you live in a area where o2 have there own cable network your line will be using bt's cabling network anyway in which case it wont make a diffrence until bt have fixed the fault. (sometimes this can take time depending on where the fault is)
as for your router causing the problem, I would try the following first check that all phone lines have filters on and test they are working, try unplugging all phone line equipment apart from the main one that your router should be plugged in and see if the router works fine.
2nd if its is still failing I would reset the router to factory default reset up the router and test again.
If that fails then you could try a different router to prove if your router is causing the problem, obviously this would need setting up with your username and password aswell.
If it is still dropping out then you will need to contact your ISP (o2) to pursue the problem, do your best effort to ensure the problem isn't on the cabling in your house as if it is if a engineer comes out it could cost a £125 charge.
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