I want to hardwire my Swann HD doorbell, rather than using it's in built battery. Can I use an existing cable which supplies 18Vac 1.25amp from a powerplug, whereas 16-24Vac 1amp is specified in the manual?
Yes, the powerplug will supply the current demanded by the bell so the 0.25A extra is not an issue.
The voltage is within the specs you gave for the bell supply.
Should work fine, just do check the polarity just in case.