The FF/REW is handled by "Mickey Mouse".
...Seriously, there's a set of gears in there that looks like the Disney corporate 'Mickey Mouse' icon, with the large circle and the two little 'ears' on top.
It's located in between the two tape hubs, on the deck plate.
The larger toothed wheel (...let's call it a 'cog') is driven by a VERY small cog on the motor shaft, and it has two "ear" cogs attached to it. The set of three ('head' and 'ears') are all mounted on a 'slippy' post, which allows the whole head/ear assembly to 'lean' one way or the other.
When it 'leans' one way, one of the 'ears' meets the toothed part on one hub; when it 'leans' the other way, the OTHER ear drives the toothed part on the OTHER hub.
The motor spins fast, spins mickey's 'head' (and associated 'ears' and the idea is that the rotating friction (which provides an opposing force) causes the whole head/ear assembly to 'tilt' (or slip) towards the required hub.
Now...
Over time, the post/shaft on which the assembly 'slips' itself gets a little sticky/gummy, and the unit doesn't engage with either gear.
The repair itself is not too complicated...
...HOWEVER...
GETTING to the repair is a pain in the butt. -There are a lot of small, fiddly and delicate parts which have to be removed on most transports... I don't know the DA-40 intimately, but I'm basing all the above on the assumption that it works the same way as all the other TASCAM transports which I do have to work on...
I just looked at a DA-98HR which is open on my bench here, and it too uses the 'Mickey-Mouse' method of engaging fast wind.
The main 'head' is hidden under a dress-plate, but you can see the 'ears, and using a toothpick or something similarly non-scratching, you can wiggle them from side-to-side. THey should move VERY freely.
If they don't, the shaft needs the 'gum' cleaning off and then re-lubricating; it's a pain in the butt to get to, and there's the risk of losing some of the VERY tiny parts, like the little 'split-clips' on top of the shafts, or one of the infuriating little spring clips (the back-tensioning arm assembly has to be dismantled to get the plate off...)
So my suggestion would be to take a look to make sure that while you were changing the belt nothing untoward got lodged in without you noticing, but if it just feels sluggish and 'gummy', then I think it would need professional attention... which may cost more than the machine is worth.
-If you have an absolute need for a DAT machine in order to transfer archived material, I probably have one which you could have for a lot less than the cost of dismantling that transport... it might be the better option.
Take a look though, and if anything differs from my description, photograph it and let me see if anything else springs to mind.
Keith