One other possible explanation would be the DIY fertiliser capsules I used when I set up the tank (using osmocote-type fertiliser). I planted these deep in the substrate (fine sand) but after adding some new plants, part of the fertiliser might have surfaced. The API test measures both Ammonia and Ammonium so perhaps most of what I saw was the latter (which is less harmful), but they both convert to Nitrate eventually (and that also spiked later on). Now things seem to have stabilised, but I will be adding the second filter just in case (I had it laying around in my aquarium stuff anyway)...
Sarpijk Thanks for the offer, I already have Pistia and Limnobium in the tank 🙂 I will be posting some new photos in the weekend.