In American English, we say: I am going to the hospital. British usage seems to be: I am going to hospital. Likewise Americans say: The Doctor will see you now. The Brits say: Doctor will see you now. This seems to happen with the words; Doctor, Nurse, Hospital, University and maybe one or two others. What happened to the "the"?

It's a Colloquialism. I happen to now live in an area of the country [Western Maryland]where people omit the words "to be" such that ~ "This needs to be painted" becomes "This needs painted" or "That needs fixed" It sounds stupid to me, but its a Colloquialism.

Brits say 'I am going into hospital' if they're going in overnight or longer. We say 'I am going to the hospital' or more likely 'I am going to (insert name) Hospital' if they are going in for a day ward or A&E visit, or going there to visit a patient. With universities, it's more common to say 'I'm at (insert name) University' (or uni) if one is actually on a course there, while 'I'm going to university' implies one hasn't started yet. The absence of 'the' with Doctor is not always the case. Certain nurses drop it, others don't. Ones that drop it tend to refer to the patient as 'we' as in 'We must take our medicine'. Also, where there is more than one doctor around, 'the' is more likely to be used than if there is only one in the place. Search me why! I also don't know why estate agents always say 'to' when they mean 'at'. 'Garage to rear' - no it isn't. It's not going TO anywhere, unless the property is on a clifftop and best avoided. It's AT the rear. Do American realtors use the same expression? Our estate agents have a language of their own.

I don't hear "Going to university" often, but we all say that we're "Going back to college."

Those cheery nurses bother me. Any how, they're substituting the title for the name so they don't have to say his name all day long. They probably address him as Doctor. When they talk to you then, they don't say "The doctor" or "Dr. Henderson", but "Doctor" instead.

I've heard 'at the univeristy' used in British English in the sense of 'at college' but it's archaic.

From Wikipedia:

The definite article

A few 'institutional' nouns take no definite article when a certain role is implied: for example, at sea (as a sailor), in prison (as a convict), and at/in college (for students). Among this group, BrE has in hospital (as a patient) and at university (as a student), where AmE requires in the hospital and at the university (though AmE does allow at college and in school). When the implied roles of patient or student do not apply, the definite article is used in both dialects.

Likewise, BrE distinguishes in future ("from now on") from in the future ("at some future time"); AmE uses in the future for both senses.

AmE omits, and BrE requires, the definite article in a few standard expressions[clarification needed] such as tell (the) time.

In BrE, numbered highways usually take the definite article (for example "the M25", "the A14") while in America they usually do not ("I-495", "Route 66"). Upstate New York, Southern California and Arizona are exceptions, where "the 33", "the 5" or "the 10" are the standard. A similar pattern is followed for named roads (for example, Strand in London is almost always referred to as the Strand), but in America, there are local variations and older American highways tend to follow the British pattern ("the Boston Post Road").

AmE distinguishes in back of [behind] from in the back of; the former is unknown in the UK and liable to misinterpretation as the latter. Both, however, distinguish in front of from in the front of.
Dates usually include a definite article in UK spoken English, such as "the eleventh of July", or "July the eleventh", while American speakers most commonly say "July eleventh" or "July eleven".

https://en.wikipedia.org/wiki/American_and_British_English_grammatical_differences#Definite_article