In the future; “going forward.” One of several cases where the British delete the article favored by Americans, the most famous other one being in hospital. Also, the football teams Blackburn Rovers and (strangest of all, to American ears) Rangers.
Wes Davis alerted me to this passage from Charles Portis’s 1979 novel, The Dog of the South. Wes writes:
An American named Jack is helping some British soldiers load sandbags during a hurricane in Belize. When one of the Brit trucks gets stuck in the mud, Jack takes the wheel, claiming to know a special way to get a truck moving again. It doesn’t work. Here’s what comes next:
Jack said the gear ratios were too widely spaced in that truck. The young British officer, none too sure of himself before, pulled Jack bodily from the car and told him to stay away from his vehicles “in future”–rather than “in the future.”
But if the President now admitted a knowing falsehood, that admission would probably be admissible in evidence against him if in future he is prosecuted for perjury.(Anthony Lewis, New York Times, December 15, 1998)/If [Mickey] Hart should ever attempt to work with dancers again in future, he should consider consulting with [Jay] Cloidt. (The Bay Citizen, April 18, 2011)
notoneoffbritishisms.com 
This seems to be one of those classic dialect markers–it’s mutually intelligible, but the grammar of it (a function-word-class item, the article) is marked by its presence or absence in the two dialects. It has always struck me as a “foreign” expression, and suggests that what we call “the future” is somehow in British English a proper noun, comparable to saying “in Bromley.” Ironically, I don’t have the same reaction to “in hospital”–while I still find it foreign-sounding, I don’t reject it to the same degree, perhaps because we may say someone dying is “in hospice (care).”
Also Bolton Wanderers, and Wolverhampton Wanderers, known commonly as “Wolves.”
I have been using “in future” and wonder if it sounds too pretentious. Again, from the Bridget Jones books and other chick lit influences.
“Going forward” is just about the most maddening recent concoction of suit-speak I can think of. I don’t care how close together someone’s eyes are or what they look like, but someone who says “going forward” just sounds like a sharp-suited crook to me.
FROM THE URBAN DICTIONARY:
“Going forward is purported to mean, “In the future” or “somewhere down the road” when in fact it is an attempt to dodge the use of these words, which generally indicate “I don’t know”. A newer development in corporate doublespeak, in most companies it is grounds for dismissal to release a press release without mentioning something ‘going forward’. Going forward, you will likely see this turning up everywhere.”
“Our company expects to make a profit going forward.”
“We don’t expect any layoffs going forward.”
“Going forward”
I didn’t realise this is a Britishism. Now I think about it, the usage is indeed strange. After all, we Brits don’t say “in past”.
“In future” and “in the future” are not synonymous. The first means “from now on”, the second “at some time in the future, but not necessarily immediately”.
I’ll do that in future = I’ll do that from this moment on.(or, at least, at the very first moment that this circumstance reoccurs)
I’ll do that in the future = not now, but one day (although who knows when?).
I’m surprised, Ben, that you can present “going forward” as a straight equivalent. Does that mean that the expression is acceptable on your side of the water? Over here (UK), it’s pure management-speak shite (if you’ll excuse the vernacular), the only useful function of which is to identify those who habitually use it as twats of the first order.
Kevin draws a correct distinction. The NYT quote seems wrong even as UK usage (assuming they meant a single prosecution). A Brit would be much more likely to say “… if in the future he is prosecuted for perjury”.
The usage in the Bay Citizen quote seems to mean “from now on”, and the missing definite article looks fine to a Brit, although I think we would be just as likely to have said “… work with dancers again in the future”.
The two comments distinguishing between ‘in future’ and ‘in the future’ are spot on: a continuing state henceforth and a potential one-off event yet to come.
Other expressions that differ are “in back”, “in back of”and “out back” in AmE, where BrE would have “in the back”, “in the back of” and “out the back”, which is strange when we both say “in front” and “out front”. Just different uses, no subtle differences of meaning.
No one mentioned “from here on in”, spectacularly ugle and thankfully equally (un)common to us all according to Ngram.