Ok for starters can we just get the nomenclature straight.
1. Just because a belt has cogs on it does not mean it is a timing belt! Yes it will be a cog belt but that does not mean that it has any function as a timing belt. Sometimes it is just a non-slip belt But that does not mean that it necessarily is used for any "timing purpose" where two shafts have to have coordinated movements. Calling it a "timing belt, if it is not used for timing may interfere with getting a correct
answer. Sometimes even the person who wrote the manual is so naive about this as to merely generate and spread confusion. ......... Cog Belt....OK ..... Timing belt ?..... Maybe!!!
2. If it really really is used as a "timing belt", unless designed by idiots, there should be some kind of indexing feature on or around the cog pulleys connected by the belt These may be as simple as small punch marks on the pulleys that have to be lined up with each other or specific marks on the machine frame etc. when the machine is assembled or the belt replaced. And the pulleys have to be securly locked to the shafts they are on by a key or pin, etc. Also any slack that is in the belt has to be taken up on the "slack" side of the belt. and this is usually provided for by a belt tensioner device of some sort. So my first question would be, "have you looked for such timing marks and are the pulleys rigidly aligned on the shafts that they fit on?".
On the cheap, but not necessarily bad, the pulley to shaft alingment might be done with a pin (e.g. a roll pin) that goes through the hub and the shaft, but in this case it is possible to get the pulley on 180 degrees out of position. The closer to fool proof approach is to have a key that fits into a slot in the shaft and into a corresponding slot inside the hole through the pulley hub. Beware tha sometimes the key is locked into place by a setscrew. And sometimes some idiot will loose the key and so just runs the setscrew in until it engages the slot in the shaft or aggggh somewhere else on the shaft. If you find that you have any of this on the pulley/shaft system on your sewing machine, one or more close up photographs would sure help especially if they are not the grainy awful photos that some cell phones take.
The fact that the machine has both a rotary bobbin and a cog belt certainly implies that it is not likely an ancient machine and may be well worth the repair issue if it appears otherwise well made.
You were a little vague about what sort of timing issue you thought the machine might have, so a bit more elaboration about what sort of malfunction it is experiencing might well help to get a better evaluation.
Best of luck, I hope that helps.