Bonus payment - which year

US citizen living in the UK. No connection to the US, except the passport - foreign employer, no W2. Bonus is paid in the UK in March (which is still tax year in the UK) for prior calendar year (2016). The bonus is paid for work done in 2016, but in March 2017. Can I recognise this bonus in my income for 2016 (the year in which I did my work) or does it have to be 2017 tax year as I am a cash basis taxpayer. Couldn't find clarification anywhere.

As you indicate, you are a cash basis taxpayer.  You received the cash in 2017.  You are taxable on it in 2017.

This I understand. However, any option for me to elect how to treat it for the FEIE purposes? It is technically for the work done in prior year (which is still an existing tax year in the country where I live).

You pay tax on the income in the year you receive it, not when you did the work.

I had two examples of this in 2016

1) I own a rental apartment and cashed a deposit check in Dec 2016 for a lease that would begin Jan 1st 2017.....the check must be included in 2016 income.

2) I billed some contract work in Dec 2016, but did not receive the check until mid January 2017....that income goes on my 2017 taxes.

Example from IRS website:

Example

You were a bona fide resident of Brazil for all of 2011 and 2012. You report your income on the cash basis. In 2011, you were paid $79,800 for work you did in Brazil during that year. You excluded all of the $79,800 from your income in 2011. In 2012, you received $18,800 for work you did in 2011. You can exclude $13,100 of the $18,800 from your income in 2012.  This is the $92,900 maximum foreign earned income exclusion in 2011 minus the $79,800 actually excluded in that year. You must include the remaining $5,700 in income in 2012 ($18,800 - $13,100) because you could not have excluded that income in 2011 if you had received it that year.

So, can you do it? Bonus is for the prior year work (which is still the same year in the country of residence - UK)

Your original question was: "Can I recognise this bonus in my income for 2016 (the year in which I did my work) or does it have to be 2017 tax year as I am a cash basis taxpayer?"

The answer to this question, is "no."

Your new question is: If I receive income in 2017 for work I did in 2016, can I (in 2017) exclude that income under the FEIE?

The answer to this question, is "yes."  In 2017, you will be able to exclude the income.