This made me wonder, those cherry blossoms must look beautiful! hope I can see them someday...

>> Maksymilian Thank you for your comment!

You can also say, If you want to tell about current, ongoing spring season: The spring has come, cherry blossoms bloom and people come to see the flowers. - but please notice that I didn't use 'When" (But I would personally use are coming as it is an ongoing action)

If you want to write about customary actions you can use present simple in a whole sentence. when valentine's day comes people usually go on dates. In your sentence you connected those 2 tenses in an incorrect way. If you wanted to use past tense you could have written : when the spring came, beautiful cherry blossoms started to bloom and people were coming to see the flowers - but it doesn't carry a recurring action's aspect.