I am absolutely 100% sure. (I can back it up with studies). This is a myth and it has unfortunately injured children in crashes.
I understand why your mum was worried about that, because it feels logical that it might work that way, but there has never been a single case of a child's neck being injured (directly) by a diagonal belt, aside from minor grazing/bruises or injuries associated with too-early forward facing in general, but injuries from improper use of the diagonal belt is a common and known injury pattern for children aged 3-8 in seatbelts, because it is very common for children in this age range to be uncomfortable in adult seatbelts, and for either the child themselves, or well-meaning adults worried about neck injury to suggest putting the diagonal part of the belt behind them or under their arm. The problem here is that under crash forces the belt rides down and essentially the child's body wraps totally around both parts of the belt, putting all the force of the crash on that one middle point, which is not good as it contains a lot of squishy internal organs which you don't want crushed, but it also puts a lot of pressure on their spinal cord, which includes the neck. A seatbelt worn properly, even if it is too large, is positioned to spread the force across their body evenly which reduces the impact of that force, as well as allowing the spine less movement, which also reduces chance/severity of injury. They actually do this very well (albeit better for adults who fit it properly!)
This video has a computer simulation of the two different seatbelt positions, showing how the seatbelt slides down when it's not held in place by the shoulder. I know the simulation has the child in a booster, but it is exactly the same mechanism of injury.
It is best for a child who has outgrown a rear facing or forward facing harness seat to be in a full high back booster, but if this is not available, a backless booster is much better than nothing, and if this is not available, then a seatbelt worn as an adult would is the very safest option, and is MUCH better than no restraint at all, or than wearing the seatbelt improperly, using items (like a cushion) as a makeshift "booster", or sharing a seatbelt with an adult.
This is not often talked about because rightly, most people say well the solution for a belt being on the neck is that the child still needs to be in some kind of booster seat. That is correct, and it is by far the safest solution but I think it is also helpful to be aware of what the hierarchy of "safer than" options is when you're in a situation where best practice isn't possible.