...A few weeks ago, Jenson started growling and snapping at the beardie. We cannot pin point the reason behind it but a lot of the time there is food or treats around, whereas they quite often share crumbs from treats left on the ground and there has never been an issue when we are giving treats to them. They can even swap food bowls and eat one another’s food and Jenson doesn’t care when the other dog eats his food. Sometimes the snapping starts with what seems to be no reason what so ever.
...Our bearded collie is a very chilled character, he freezes whenever Jenson has snapped at him.
... The vet seemed to think that despite puberty, it was very uncommon for Jenson to start attacking a dog with whom he has lived with since being brought home.
...as Jenson sometimes gives no warning that he’s about to go for the other dog and so we have no time to grab hold of him. We worry that keeping them separate may build up more frustration in Jenson.
For me, the above excerpts of your post contain a LOT of red flags and I think that you are absolutely right to be concerned. Several quick points. First, I agree with the suggestion that you get a balanced, experience trainer on board ASAP. If you post your location, forum members may be able to recommend one or more possibilities.
Second, I think you've missed or misunderstood several signals that things haven't been going as smoothly as you believed (see excerpts above). Understandable, dog behavior is often subtle and very quick; that's where a trainer can be particularly helpful. For example, your bearded collie may have been "chill," but he also may have been increasingly intimidated by a pushy puppy.
Third, I disagree with your vet's assessment. It's not at all uncommon for puppies to start sorting out their position (aka trying to take over) especially when they hit puberty --- though I suspect that the puppy started asserting himself prior to that (e.g., that "sharing" that you thought you saw was him TAKING the collie's treats, etc). That's not something that neutering will "fix" and it may well make the situation worse.
Fourth, I agree with the recommendation that you remove all toys and feed and treat separately NOW. Feed both in their respective crates, so that they both feel safer. As well, crate the puppy when you're not there to supervise them directly and keep the pup on a longline (so you can correct rudeness immediately) when you are. Again, get a trainer to help up your game.