Masters degrees can be applied for from October onwards. The big recruitment drive for them is between November and March - after this time you may find that some popular courses are full.
PhD recruitment also takes place throughout the year as and when academics are informed as to whether they have been awarded a studentship. PhD places are advertised right through to the end of September, but you should try to have something arranged by May as after then the choices become more limited and competition can become much stronger. If you wish to fund yourself through a PhD you can approach academics at any time of year and will be able to start at any time of year.