This phrase means "we are on time" - present tense.
"I hope we will be on time" is a future wish.
Past tense "Yes, we were on time (luckily)!"
Your context is not clear - "I hope we (will) arrive on time." The word "will" has been left out, which without proper context makes it unclear.
Thank god we’re on time sounds more emphatic to me than « heureusement nous sommes à l’heure ». Thank good, to me is a very commonly used exhortation, used for emphasis and I’d say almost never has any religious connotation.
"Heureusement nous sommes a' l'heure" - means - Luckily we are on time/ Luckily we shall be on time. It is not clear which of these is true - present or future - only context can tell - also possible I hope we are arriving on time/ I hope we shall arrive on time/ Luckily it looks like we are going to arrive on time. Only the person on the train can say which is correct - context/contexte.
Thank God (rrose17) (not good) is sacriligious to many religious people = (blasphemy) but is SO common that actually you are right!!!
However He is usually God with a capital "h" H and a capital "g" G since He is the one true God (apart from all the others!!!!).