Well, you could either buy a 'lot' of tea, or gather 5 or 6kgs of oak bark, and boil them up with that......not all that cheap to be honest for the tea, or the fuel unless you're somewhere woody
or,
buy one of the wash in dyes meant for just what you are trying to do.
http://www.thedyeshop.co.uk/
Just make sure it's a dye that clearly says that it will dye cotton.
No connection with the company, just a happy customer.
The dye will come as a little sachet that you just put into the washing machine after you've washed the trousers without adding any fabric conditioner.
Wash again to wash the dye in, wash again to remoce excess, hang them to dry and wash the machine out ( I usually use this last wash to do the cat's bedding) again to make sure there's no dye left.
Result
The problem is that cotton is a cellulose fibre and it's hard to get natural dyes to chemically bond with it. Wool is a whole other story, but cotton, linen, hemp, ramie, nettle, are all white, brown, beige or yellow for a good reason. That said, indigo (blue) a substantive dye works really well, that's why jeans are the colour they are
The nylon stitching might not end up the same collour as your trousers, but it should still take up some of the dye.
It's not really that nylon doesn't dye, it's that it doesn't take anything much apart from yellows, peaches and blues from the natural dyes, indigo is expensive and complex to work with effectively, and I couldn't see you wearing peach coloured breeks
cheers,
Mary