In order to warp, your ship has to be headed in the direction you want to go at 3/4ths it’s top speed.
The align time on your ship is how long this would take from a dead stop under it’s normal engine power.
If you use a MWD, when you turn it on your maximum speed goes up, and your acceleration curve is adjusted upwards to match. (ie, it takes the same amount of time to reach top speed), but when the MWD shuts off, your top speed suddenly drops, but your ship’s current speed remains the same as when it was on. The end result is that you end up mathematically at 75% speed when your maximum speed suddenly drops.
Also take note that being webbed will trigger the same sort of effect. If you are trying to warp out and someone webs you without applying warp disruption, you might be able to enter warp much sooner than you normally would because your maximum speed suddenly drops. People commonly apply this to freighters to get them into warp very quickly.
Nanos improve your agility, and so improve your acceleration curve, but you’d have to be in a pretty specific band to need them for the MWD trick. You’d have to be just shy of reaching the required 75% speed without them.
Because ABs cycle slightly faster, I will first see if I can get away with using one of those, then move up to a MWD if the AB won’t work. The nanos may help if you have some velocity in a direction not in line with your warp target, though. Higher agility would equal a faster course correction.