As you state, a three month notice requirement is unusual for part time workers (or for workers in general who are employed in relatively low-skilled occupations). In practice many companies would not seek to enforce such a period as long as they'd found someone to take over the role. (However 3 months is far less than the notice period required for people in senior positions. Many workers in the finance sector have to give 1 years notice. Some consultant surgeons have to give two years).
As Factor30 indicates, there is (for adult workers) no right to any break whatsoever unless a shift exceeds 6 hours. (If a shift is exactly 6 hours long there's still no statutory break entitlement). Then there's the right to a single 20 minute (unpaid) break, irrespective of the total length of the shift. (e.g. if you work a 24 hour shift the entitlement is still to a single 20 minute break, not to several of them).
However periods of sickness may not be deducted from holiday entitlement. Even if someone becomes sick while they're already on holiday, they must then be given additional holiday entitlement to compensate for their lost holiday.
http://www.direct.gov...dholidays/DG_10034711
Chris