'Ad-este' means 'be present', I think 'O come' is quite a fair translation of that.
I think it's a splendid hymn in both languages - it's not easy to find a good recording of the Latin version on YouTube,
I was looking for one yesterday - the performance in St Peter's is most operatic:
The origin of the hymn is obscure, but the earliest document with the tune and the first four verses
(in the order you've given them - they vary, as in the Vatican version above)
is from the English Catholic college at Douai, c1740. It was first heard in England
in the chapel of the Portuguese embassy, and the Portuguese claim it was composed by one of their rulers, I forget which.
For a time it was only sung by Catholics, and was even regarded with suspicion as 'Jacobite',
but the 19th century Tractarians (Anglo-Catholics) adopted it,
Francis Oakley who translated the verses that are usually sung
was vicar of the very 'high' Anglo-Catholic church, All Saints, Margaret Street, in London.
Just read this carefully and have a bone to pick. My statement that adeste means arise was wrong because it was a typo. I meant arrive, for be present. But I don't think in Latin it is the same as come. It is one of those words in Latin that reflect more my specialty, the military side. It comes from adsum , meaning (lit.) I am to. Remember, the military usages in Rome were ubiquitous. Many things and words must be interpreted in terms of how a soldier would use them. Adsum, Present Sir, is a roll call response. The person is not coming, but is there. So Adeste Fideles, plural imperative (from Latin for a military command) is freely translated as "You Faithful ones had better be there!"