Redundancy is diminished requirement for the work in question so it's not a redundancy situation. Employers used to say there was a diminished requirement for the part time position because the increase in workload required a full time position but that approach is now considered risky and not good practice.
My advice to an employer wanting to increase someone's hours is to approach it as a terms and conditions change.
Ahh bugger. It's actually a friend who works part time, has done for yonks (3 or 4 years) but bosses are getting shirty about it and asking when she will be returning full time.
It's my understanding that she doesn't have enough work to do as it is, and they're not nice to work for so full time really not within her interests. She was hoping to be able to request redundancy if she turns down the full time role.
That wouldn't help her. She can refuse to agree to a change in her terms and conditions, so arguing that there aren't any terms and conditions would help them not her.
Evading signing a contract isn't an achievement, by the way. A contract is helpful to both parties. She will need to argue that her current working hours are her terms and conditions and therefore cannot be changed without her consent. If she's been working those hours consistently for a long time, then they are her terms and conditions, but obviously if she had a contract with those hours specified, that job would be easier.
As it is, with no contract, her employer might argue those aren't her terms and conditions anyway, in which case they don't need her consent to change them. Its not that they'd (necessarily) be successful in that argument, but it's easier all round if there's no debate about what the terms and conditions actually are.