Hi, how long are the queues to purchase tickets and tobaggan ride at Beijing Mutianyu Great Wall?

Well, it is hard to say. And the situation would be different during the high peak season. Thus you are suggested to book the entrance tickets in advance. TravelChinaGuide.com provides this ticket booking service. Thus you can ask help for them. If interested, their contact information is bookingtravelchinaguide.com. :)