In my first week on HRT (mirena coil and 25mcg Estradot patch) I had exactly the same. It's horrid. I hate chewing gum, but resorted to that for a while, but I was getting increadibly bloated due to swallowing air. I also noticed that my mouth was a lot drier than normal (it woke me up a few times). 2.5 months later it's a bit better in that I don't have it all the time, but I probably experience it at least every other day, especially if I've been talking a lot. I started getting saliva stone a few years ago, so have been massaging my saliva glands to increase the saliva flow (there was a short video on 'heal with Samantha' on youtube), and this has definitely helped. I've also got an artificial saliva spray to use if I wake up in the night (it's more refreshing than water). Some people use xylitol (you can add it to water with baking soda to make a more refreshing water), or xylimelts but I find it also upsets my stomach a bit. I've also tried to have more sour foods to help stimulate saliva (e.g. grapefruit, lemon).
I have a mixture of other symptoms which to me suggest Sjogrens syndrome, but GP thinks possiblly lupus, so I'm waiting on my appointment with Rheumatology to investigate further. In the meantime I've been reading about auto-immune conditions and it seems that they are often far more common in women and so likely to be hormone related. My GP thinks it's a coincidence that my symptoms flared after starting HRT, but we don't really know. If nothing else is helping, maybe ask if you can have a blood test for an auto-immune condition?
Good luck with your investigations. Hopefully you don't have an auto-immune condition, but the saliva gland stimulation (massage and sour food / drink) might help? Please let us know how you get on. It sounds trivial, but this metallic taste is a symptom I really hate as it's hard to ignore and makes me a bit of a glutton (eating and drinking to try and take the taste away)! Which reminds me, try to avoid anything with sugar (e.g. boiled sweet) as it might cause mouth ulcers, which just makes everything worse.