"The ambulance took my husband to the hospital."
Translation:Thóg an t-otharcharr m'fhear céile chuig an ospidéal.
Because of the preposition chuig. Like other prepositions, chuig + an would normally cause eclipsis, but in the case of nouns that start with a vowel, you just get the plain, unmodified word, without the t-prefix that ospidéal would get in the nominative case after an.
If you used 'go dtí' instead of 'chuig', would 'ospidéal' get the 't-' prefix?
Yes it would - go dtí is a little bit odd, in that it doesn't trigger the dative (as the simple prepositions do) or the genitive (as many of the compound prepositions do).