Hard to advise without seeing and hearing for myself. But things to check....
-pipes secured properly
-Is it mains water fed or tank fed...try winding the stopcock back half a turn... it shouldnt be open fully open and may be a pressure problem.
-bottom fed and side fed systems reduce the chances of water hammer (especially bottom)
actually having just reread your post and seeing its happening in both toilets you have and next door, I would imagine its related to water pressure. try turning the water off altogether and then flushing with the water off.... see if that makes a difference to rule it out..... then turn it back on and see if it happens on refilling. wind the stopcock back a bit.
It sounds like a rubber seal in one of the stopcocks is vibrating as the water passes though it (same principle as the reed in a saxophone)
the stopcock needs a new valve seal and/or is not fully open...or the toilets.... if its the old float valve systems they are very prone to it.... just change the diaphram valve.
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