How long does it take for a tribunal?

Here's a bit of a back story, in case anyone has any tips (I have and will continue to read up in here, so thank you), but skip to the end for my question if you'd rather not read it all!

My soon to be ex wife and I returned to the UK from living abroad (where we separated), gave a tenant who was living in our jointly owned property notice and put our property on the market. She then decided she no longer wanted to sell it and moved back in, against my wishes. She is now living in and refusing to sell our house and is trying to revoke my 50% access to the kids, which I have had since we separated as agreed between us (I have emails stating this but nothing formal in place).

I firmly believe she is doing this so that she can receive child benefits to continue to be able to afford to live in the house.

My ex threatened to go to the CMS unless I agreed to make her monthly payments and emailed me a suggestion on it, but then I found out that she'd gone to them anyway behind my back.

She told the CMS that I have the children 'up to 174 nights', which was a lie.

I applied for "shared care" through the CMS website a few weeks ago, and on Monday logged into the site to see that it has been rejected. At the time of applying I uploaded a letter stating:

• I will be having the children overnight 183 times this year whilst my ex will have them 182 (just the way it falls, but it is a fact).
• I am the main point of contact for the children’s school and have had to collect them due to illness on two occasions since my ex and I split. She is unable to collect them during school hours so I would always do this. 
• The only time they have visited a medical establishment since we split, I took them.
• I drop the children at school and collect them from school more than my ex does, because she uses child care (non registered).
• Because my ex is a teacher, I have taken time off work on several occasions (and will continue to do so) to look after the children on Teacher Training days.
• I have made a higher monetary contribution to the children’s school uniform. I have also paid for school trips and special dinners (such as at Christmas).
• The children have access to their own bedrooms, clothing, books, toys and games, games consoles etc at my house.
• The children do the majority of their homework when they stay with me.
• My ex earns more than me (by a minimal amount, but still).
• My ex has refused to enter into the mediation process to discuss child care or financial matters. I have attended a MIAM session.

My ex is receiving child benefit for both of our children, because she refuses to allow me to have half of it.

The reason for the rejection has now come through as "Both parties are disputing level of overnight care no formal contract such as a court order has been
provided. Shared care to remain as band C".

After various calls I have today spoken to my "case worker" (I seemed to know more than she did simply by what I have read on this site), who has told me that there's nothing I can do as it is her word against mine and I will need to appeal to a tribunal. I have a c100 form that states that she has refused mediation so am hopeful of obtaining a court order that will state that I have 50% shared care, but I'm also not holding my breath given my experience so far.

MY QUESTION: I asked the cms case worker how long a tribunal would take and she stated "the average time is currently 67 weeks". I was obviously shocked at this, but am now wondering whether she might have said 6-7 weeks? A Google search suggests that it will take 6 months, which unfortunately suggests that it was the former, but to save me wasting my time calling them again, does anyone know whether it will indeed be 67 weeks? If so, and I appreciate that others have it worse than me, it will cost me over £5k in payments before any decision is even made...all based off a lie!

She also said it would be up to me to get those payments back.