I have two old Wilex fuse boxes fed from two RCD's and it is common when one RCD trips when resetting it the other one will trip. I will assume switching causes a spike which trips the other RCD? Same when fuse box switched on with main switch, often trips RCD, but switch on one MCB at a time it's OK. The RCD's are around 1996 and no electronics the modern RCD is far better.
However in this case think we are looking for a neutral earth fault. As said since many switches only switch the line, switching off does not help. Also I have found even going around with a multi-meter the 9 volt or less used by the meter often can't detect the fault. My insulation tester went missing, however found I could get a cheap replacement at £35 so not too bad.
As to DIY well anything which can unplug then easy to test, but if it will unplug then no real need to test, just unplug it. Since the fused connection unit (FCU) with a switch often the switch is double pole then again switch off, no real point removing fuse.
Although we have earths to lights, most bulbs do not have an earth connection, so first thought is no point removing bulbs. However experience says do remove bulbs, I have found a bulb full of water, seems bath had leaked, leak cured, and lights left off for 3 months waiting for me to come home. (Working in Falklands at the time.) There was a hair line crack and bulb was full of water. So yes remove all bulbs.
If this still does not allow the kettle to be used then it likely needs some disconnection inside the consumer unit to locate fault, I personally feel if that is required it needs some one with skill, and test meters, it is beyond DIY. Since the main switch turns off both line and neutral with the main switch off it is quite easy to test all neutrals to earth, so likely one at a time removing and testing neutral wires is not that long of a job. Once you know 100% which circuit however, the tracing can then become a slow process.
I had it with daughters house, one by one I was moving socket to socket isolating parts of the ring, it turned out the fixing screw in one socket had caught the neutral, in the bedroom, and just walking in the room would cause it to make and break contact so lucky to find it. Also seen where spiders or mice have caused the problem. It took me around 3 hours to find the fault, it's just luck as to if first or last socket removed has the fault. But it did need the meter to find it, and so I would say once all unplugged and all bulbs removed, then time to hand over to an electrician.