Lidl in Germany has an offer on this week for a wash-in impregnating agent. This may turn up in the UK a bit later. Look for the "Trekking" theme. There is a spray and an impregnating agent to use in the washing machine.
I don't know if it would be suitable for the coat you describe though. Perhaps you could ask a dry cleaner?
What kind of rain coat are we talking about here? If it is of waterproof cloth it should only need to be washed on a gentle cycle then re-proofed. Sports' shops sell a liquid waterproofing that only needs to go in the rinsing water.
If the coat has a synthetic rubber backing it may not be possible to get rid of the smell, but it might be worthwhile to mention the matter in the shop where you bought ti.
I have Seasalt raincoats for summer and winter and the recommended cleaner is Nikwax. I bought it on Amazon, it's 2 bottles, one to wash and one to re-waterproof. The bottles will last 4-5 uses so worth the expense. Seems to work!
I have some Waterproof Clothing Wash from Waitrose (Waitrose brand) which I used to wash a rain jacket from Seasalt. Seemed to work fine. I found it in the detergent aisle.
I wash mine in the machine on a delicates cycle in a small amount of washing-up liquid put in the machine drawer. Not too much or folk will think you are having a foam party. I then hang it on a coat hanger either inside or out, depending on the weather. As other posters said, you can by the spray to re-waterproof it. I do that outside as sprays affect my breathing.
I have a White waterproof winter coat and wash it regularly. There are two bottles (from Amazo) one to wash the coat. The second to do a cycle to waterproof the coat. Works great. Bit expensive at about £15 but well worth it.