I've been in that position more than one when renting privately, working and no benefits. It is gut wrenching but I just looked for somewhere else to rent. But is that their circumstances?
However, in the short term, the obvious course of action is simply to seek alternative accommodation. These links might help:
http://www.primelocation.com/to-rent/
It might also be sensible for Dad to have a word with his boss to explain the situation. Then, if he needs time off to view a potential new home, his employer might be sympathetic to his needs.
I think it's standard to have to give 2 months notice from both sides.
It is much easier nowadays with the internet. Start looking at all the local estate agents and nose round the available houses on line. She only needs to view the houses she's interested in and not the ones the agent suggests.
I know it's hard work with little ones but it's the same for buyers as well. I sold my house and the buyers wanted a date. I had a similar amount of time to find another one and I had 3 little ones.
Moving house is stressful regardless if you rent or buy but I do appreciate it'll be much more stressful when it's a forced move rather than choice.