Any decision takes immediate effect.
So, if you think they may reduce it, you want to apply as late as possible, but not so late that there'll be a gap, but if you think they'll increase it, you want to apply as soon as possible.
In theory, so long as you've kept the DWP fully informed of any changes in your care/mobility needs, it shouldn't make any difference, since you'll continue to get whatever rates you're on now, but given the vagaries of DLA decisions, it's a gamble.