*physicist hat on* whilst wormholes and the Alcubierre "warp" field do exist as solutions to GR, it's pretty much certain that there's no way to use them for practical ftl travel
First, both solutions require some profoundly weird stress-energy tensors that don't behave anything like any known particle, field, or combination thereof that we know of (even including only partially understood things like dark matter and dark energy). This is why you often hear talk of "exotic matter" when these things come up, but "magic matter" would get the point of how implausible this is across better
Secondly, and more fundamentally, there are topological issues
In the case of a wormhole, there is a topological difference between a universe with a wormhole and one without (and between one with one and one with two etc), such that there is no way to continuously deform one spacetime until it looks like the other. This means there's no plausible way of making new wormholes
In the case of the Alcubierre "warp" field, on top of similar issues around having to change the topology of the universe, the interior of the bubble is topologically separated from the exterior. This means that there isn't any way to move something from outside the field to inside it, meaning that even if you could make the field (which you probably can't) you won't be able to put anything in it to send faster than light
Thirdly, whilst quantum gravity is definitely not anything like a complete or coherent theory, there are some problems in it that are soluble. In particular, people have done some calculations around the self-energy of a wormhole, and found that it would provide a strong pressure towards the two ends becoming as close as possible and, if possible, collapsing to nothing. Given the intense fields in an Alcubierre solution, it seems very likely similar effects would take place there due to Hawking radiation (which would likely cook anything you did manage to put in there)
Fourthly, and this isn't so much a physics problem as a philosophical one, any ftl system can always be used to violate causality. No matter how smart you try to be with arranging your ansibles, it is always possible to set up an arrangement of observers such that later on, a message can be sent back to before its composition (i.e. a tachyonic antitelephone). This isn't a problem physically, but it's an unavoidable consequence that isn't normally considered
And lastly, if you did somehow get something moving faster than light (this is also not a problem per se, the issue physically is mainly with things accelerating from below the speed of light to above it, or vice versa, which would require infinite energy) you have to deal with the rather counterintuitive dynamics of tachyons
Namely, they lose energy as they accelerate and will constantly emit Čerenkov radiation (definitely gravitational, and, if charged or magnetic of the more usual electromagnetic type) which occurs whenever an object carrying a given field's charge moves through a medium faster than the speed of the force in that medium (i.e. an electron moving through air faster than the speed of light in air, or a dense object moving through air and forming a Mach cone), which together mean they will naturally tend to accelerate to infinite speed unless constantly trying to decelerate
The faster the object goes, the less time (in its weird reference frame) it spends near enough another object to interact with it, and so the lower the likelihood or effect of any interaction so any ftl object would rapidly stop interacting with the ordinary sublight universe
All in all, this means that it almost certainly isn't possible to have an ftl system, even more certain isn't possible to make one, and if you somehow did, it would be inherently unstable. But you could use it to cheat the lottery
