Those mains rotary tools are pretty generic being the same or similar to each other with the Argos unit now branded under the 'Guild' name.
I had a similar unit that I stupidly 'lent' to someone and it came back with a similar problem of not being able to lock the collet chuck as the pin wouldn't engage. On stripping it apart the pin was destroyed along with the spring and the slot that it locates to on the motor shaft. (Stupid design!) I have a feeling they had 'accidentally' engaged the locking pin whilst the damn thing was still running.
I never did find spare parts for it or rather did in the end but it would've cost more than a new generic tool so I purchased another online. This time however I also purchased the rotary shaft that fits onto them as they have a 'holed' system to lock the chuck/shaft so far less likely to be further damaged.
Needless to say I've never lent this one out!!
The older unit sits in a draw just in case it comes in useful for spares as the switch, motor, brushes, etc. still all function.
I hope you get sorted or at least able to purchase another.
(My online version cost far less than the shop bought one but is exactly the same!)
All the best,