Turn on the lamp LED late to enter the voltage

An Arduino can be programmed to give any delay.

Nano would require a 5V supply - even adding a buck convertor board would be the cheapest option.
https://www.ebay.co.uk/itm/133966923257?hash=item1f310d05f9:g:IC4AAOSwxlJfj1bN
https://www.ebay.co.uk/itm/19359137...0NQ7n2MsNjQrO28FQTcB9veVLwKk|tkp:BFBMqNKKicBf
Uno has built in regulator so could operate from 12V supply.
https://www.ebay.co.uk/itm/265366521662?hash=item3dc913c73e:g:WKUAAOSwDQdhbX0z
Note the above are random sellers picked only to illustrate what is required and not recommendations.