"Heddiw, dw i wedi mynd."
Translation:Today, I have gone.
Yes, but wedi is more present perfect, as I understand it, than simple past. As in "Tom is not here - he has gone to the party already."
There is a separate simple past tense in Welsh which would be closer to "I went".
That said, using the present perfect together with a time ("today, I have gone") sounds very odd to me in English as well.
I think here the problem is not the time, because 'today' is an incomplete time period, so it's OK in English to use present perfect (e.g. 'Have you seen Mary today?'). Rather, the problem seems to be with 'gone' which in English means that you are no longer present (e.g. 'Where's Mary? Oh, she's gone to the beach today.') It seems to me that the only time we would use 'I've gone' in English is when we leave a note for someone who expects us to be somewhere explaining that we are somewhere else (e.g. (I've) gone to the supermarket, back soon.) Even if we phone to explain our absence we wouldn't say, 'Hi, I'm not there because I've gone to the office.' We'd just say, 'I'm at the office,' or 'I've come to the office' (because the movement was in the direction of where the speaker is now, so we use 'come' not 'go'). English is also strange in using the past participle 'been' to mean 'gone and returned'. (e.g. Are you going to the bank? No, I've already been today.)
I think there is a difference across the Atlantic. Compare British, 'Have you ever seen that film?' and American, 'Did you ever see that movie?' (and in Br.Eng. 'ever' is a very strong present perfect marker.) In British English incomplete time periods are usually 100% compatible with present perfect, as they bridge the gap between past and present.