I am a mere woman who has no idea but it should never have got so bad . Lazy husband eh! . I am a firm lover of super glue for everything that needs fixing. After you have glued all the stray bits just get some outdoor material of your choice and make some arm rests with pockets for the cold ones.
Welcome to the Bunnings Workshop community. Thank you so much for joining us and sharing your question about how to repair your outdoor chairs.
It's great that you've received excellent advice from @brendawebster44. I would take her suggestion and just take it one step further. After you've glued all the loose strips back together, I recommend building a timber armrest that will protect your weaved chair from further damage. I've drawn an example below.
You may build it out of an 1800 x 450 x 18mm Clear Pine FJ Laminated Panel. The example I've placed only goes outside and is not fixed in any way so that you can remove it and give it a wash. Please have a look at the drawing and tell me what you think. If you wish to see it in another configuration, please let me know. I will be more than happy to re-draw it for you.
If you need further assistance, please let us know.
I agree with Eric. This looks like the easiest solution really, if you intend keeping them. I certainly can't think of another option to repair them, so disguising the issue is easily the best.