“Don't try the paranormal until you know what's normal.” - Granny Weatherwax
Some class cards say, that they give an ally a bonus for their attack value if they are adjacent. When does this occur? At the time the card effect is execute or is this an ongoing bonus and my allies get the bonus as soon as they enter a space adjacent to me?
I think you have in mind cards like the Tinkerer's Flamethrower (ongoing defense bonus on himself and adjacent allies) and Enhancement Field (ongoing attack bonus on himself and adjacent allies). Those card (the lower half) have a specific icon that identifies their effects as round bonuses and they will be continuously active until the end of the round.
In these cases the effects are like a movable aura continuously centered on the Tinkerer. The card itself (most notably, the fact that it has been played in the active area on top of the hero sheet as a round bonus) is enough to keep track of the movable aura.
If it was otherwise, some bookkeeping token should be assigned to the affected characters at the time the card is played, but Isaac's design is strongly focused on avoiding fiddly bookkeeping during combat.