I can answer pretty well for the state of things in the UK.

Were they able to save enough of their wages to provide for retirement?

Generally speaking, yes. Their wages were mostly terrible, but they would presumably move up in pay/position throughout their career. There were books written specifically for servants, one of the most popular being The Complete Servant by Sarah and Samuel Adams, that explicitly encourage retirement saving and even include tables of compound interest rates for planning purposes.

Also, while there was rarely a formal agreement, it was a common practice to make a modest provision for long term servants in one's will. This would amount to a pension/retirement income. This would not have applied to young maids and footmen, but governesses/nurses who stayed on to care for multiple children, butlers, housekeepers, stewards - yes.

illness or injury

Temporary illness or injury in employees is something to be expected. If someone had to stay in bed for a few days, everyone else would just pick up the slack. If a doctor was necessary, this was provided at the employer's expense. Although we see a lot of cold indifference from masters to their servants portrayed in fiction, there was a good amount of literature/pamphlets/aphorisms promoting a healthy, happy household - of which servants were very much a part. Employers were perceived as being responsible for their servants' moral and physical well-being.

There were two ways to employ servants; year-to-year, or at will.

Year-to-year servants had a contract that could only be broken by mutual consent or with three months' notice on either side (or three months' pay in the case of an immediate dismissal). They could not be fired as a result of illness or injury, nor could their pay be withheld. The employer could give notice that he would not be renewing the contract at the end of the year, but the servant was entitled to his wages for the term of the contract, even if he couldn't physically work.

At-will hirings were more temporary/seasonal positions (like staff hired on while the family was in London for the summer) and only required one months' severance/notice to leave. These people were a lot more vulnerable, but their weekly wages tended to be a little higher on average.

How would they find another job?

Via newspapers/magazines, or an employment agency, or by direct hire/reference. People would also absolutely poach good servants from one another. It was terrible form, and would ruin friendships, but it happened.

Would a year of work pay for a year of not working?

It depends on the position held and how a person is willing to live while they're not working.

What we would consider a lower-middle-class income might be somewhere in the range of £50-100/year (ca. 1825). If you're a governess coming from a household in this income bracket (as many were), you'd probably earn ~£30/year, so obviously you wouldn't be able to afford what you're accustomed to on your own... but could you rent a room somewhere and buy food for £30/year? Probably, if you were very careful, and had almost no unexpected costs.

Understand though, that a governess is one of the highest paid positions in the house. She's arguably not a servant at all - more something in between.

Here's a sample list of salaries for comparison: Butler - £52 Housekeeper - £25 Nurse - £21 Maid - £15 Footman - £24 Scullion - £10

The average unskilled laborer was making £10-20/year, while a skilled laborer made about twice that, which is still not much. These people managed to eek by with families even - though other family members would probably also work. It's not impossible, but it's a pretty meager existence.