The original country laws by default, but that can change.
Once a country(let's say Elbonia) takes unconstested control of a town/region(let's say Silvania), it becomes an occupying power and has the obligation of ensuring the safety and well-being of its population. It usally also has the practical need of ensuring that the zone does not plunge into chaos, as it is usually detrimental to using that zone resources to support the war effort and may force the occupying force to deploy troops needed elsewhere.
To that effect, an Elbonian occupation authority is established and usually a proclamation issued and distributed informing the inhabitant of the new authority and its decisions1. While by default the original laws will be in effect2, the occupying force has the ability to issue new laws and to enforce them in order to conduct its activities.3
The paramount example would that of treason: a Silvanian citizen caught acting against the Elbonian occupation forces can be tried and punished by the Elbonian authorities, even if Silvanian laws allowed or even asked for Silvanian citizens to attack Elbonian forces.
At a practical level, if Elbonia follows the rules of war then the occupation of the zone is temporal (as wars of conquest are forbiden) and it is usually easier to let civilians handle themselves as long as it does not affect the war.
In any case, a situation of war is often traumatic enough that even without occupation its effects will be felt; in your example Silvania's government could just decree a 90 kph speed limit in order to save fuel, establish fuel rationing, or even require government authorization to use highways in order to free them for military transports, setup checkpoints every few kms...
And of course, remember that for some people inter arma enim silent leges.
1There are some explicit prohibitions against laws forcing the inhabitants to directly support the occupying power, like forcing the civilians to enrol in its army or to build military equipment.
2Art 64 of the Fourth Geneva Convention:
The penal laws of the occupied territory shall remain in force, with the exception that they may be repealed or suspended by the Occupying Power in cases where they constitute a threat to its security or an obstacle to the application of the present Convention. Subject to the latter consideration and to the necessity for ensuring the effective administration of justice, the tribunals of the occupied territory shall continue to function in respect of all offences covered by the said laws.
The Occupying Power may, however, subject the population of the occupied territory to provisions which are essential to enable the Occupying Power to fulfil its obligations under the present Convention, to maintain the orderly government of the territory, and to ensure the security of the Occupying Power, of the members and property of the occupying forces or administration, and likewise of the establishments and lines of communication used by them.
3Art 65 of the Fourth Geneva Convention:
The penal provisions enacted by the Occupying Power shall not come into force before they have been published and brought to the knowledge of the inhabitants in their own language. The effect of these penal provisions shall not be retroactive.
This answer the "how would one know if one has been conquered?" part of your question.