I teach in an 'immersion school' in Spain. The children in the infants will begin to write their name whenever they do it in Spanish. Remember that writing really should follow on behind speaking and reading as it usually does with native tongue. It's a case of whenever the children are ready, rather than a specific age. As this is a 'school' and the children go through, each teacher will be assessing the children when they arrive in the class. I teach year 3, 6-7 year-old on arrival. All of them have started writing, but their performance will vary greatly given their maturity, ability etc.
PS, teaching the alphabet is crucial. 1. If the children are going to learn to spell, they'll need to know the letter names.
2. It's not true to say that English is not a phonetic language, it's just that there are a lot of irregularities. Most of the irregularities occur with the vowels, not the consonants which are more regular. The key to teaching English spelling is to teach the regularities and the irregularities. It isn't easy. But it kan bee dun. (Whoops!)