Well, there would be resistance at first. We might do pretty well nuking them at he start, but when the SM come down, they'd probably decimate a major country or something. In the worst-case scenario, either we go all up against them and they declare us heretics and spawn of Chaos and Exterminatus us, or just denounce us as heretics straight away and Exterminatus us. Assuming this doesn't happen or they don't Exterminatus us, (they'd probably recognise the galactic coordinates of Holy Terra, and therefore want to preserve it) then any long wars would be doomed to failure. The Imperium has 1,000,000 Space Marines and hundreds of thousands of millions of billions of trillions of Imperial Guard. This wouldn't be a war we'd get out of easily, and the situation here would be like on Cadia in the sense of these two quotes:
"The birthrate and recruitment rate on Cadia are synonymous..."
"Any child on Cadia who can't strip, reassemble and fire a lasgun before he can read was born on the wrong planet."
Even with this, we might end up barely coming out, not just because of the Imperium's far superior numbers but also because of CREEEEEEEEED! (Though it is possible we could match Creed, since modern tactics are far superior to 41st Millennium tactics and our tacticians would be tactical geniuses by 40K standards.) However, most of the IG will be up in their ships for most of the time so a few nukes and missiles into the Imperial Navy could tip the balance in our favour.
Eventually, we would probably still end up losing (damn you Space Marines and your overpowered statlines and special rules!) and we would have to accept the God-Emperor as our rightful saviour and protector before he's even saved us. Life as usual, 40K style, and we both contribute something to the other. The Imperium gives us Warp travel and the opportunity to ply the stars, we give the Imperium better tactics and some scientific innovation. A couple of revolutions will spring up, and I know for a fact that one will end up being quite large and then execute Robin Cruddace and Mat Ward for making IG and SM so amazingly overpowered. Others will probably have someone come in and say they're actually the Emperor, since for all the Imperium knows they could be as this is the distant past and the Emperor is probably among us somewhere. These rebellions would fail whenever they sprung up.
In a few millennia we would get up to the Dark Age of Technology, as our top scientists and the Adeptus Mechanicum finally manage to reverse engineer the weapons and technology of the Imperium and use this knowledge to pioneer the things that were left behind and not yet found in the 41st Millennium. Thus, we enter the Dark Age of Technology and we encounter the races that also exist in 40K. We reign supreme for another few millennia, then become like the Eloi from H.G. Wells' novel The Time Machine, or possibly more like the humans from Wall-E as our great technologies drive us into decadence and laziness. We fail to notice the encroaching threat of the xenos and Chaos, and then we are dealt a great blow. We enter the Age of Strife, and there would have been rebellions going on for all this time. At the end of the Age of Strife, one man is successful in his claims that he is the God-Emperor. By this time, many have forgotten that he existed, and the rest is history.
Thinking realistically, we see no threat in the large armies of 30cm tall metal and plastic miniatures, and after some petitions to the governments of the world by wargamers and hippies, we allow the Imperium to live, and we get one of the High Lords or Tigurius to tell us where they came from and how they got here. We then manage to get all the other armies out of the 40K universe, and after some persuasion (e.g. capturing major gods and leaders and keeping them in underground facilities) we get the armies to submit themselves to the service of wargamers, dividing them out between us. A rewrite of 40K is then put in place, a cloning system based on Dark Eldar technology is developed to make sure miniatures which get slain in your battles can come back, and also so that armies where there are not many warriors to go around can be given out to everyone. And 40K is now a lot more awesome.
If the Imperium is at real scale, and they're accepting diplomacy from us, then after a few weeks, maybe a month or two to give our answer to the Imperium's ultimatum the governments of the world send wargamers who play Imperium armies to go up and send our message and to swear oaths of fealty to the God-Emperor on behalf of the human race. In these few months, every wargamer has wisely switched to an Imperial army and we become the world leaders and generals in a new Golden Age of Humanity. 40K is reinvented as a means to test out tactics for battles and campaigns against xenos.
This is a warhammer forum. Everything is a military operation.
I have spent the last 25 minutes doing reconnaissance on my kitchen for operation: Cup of tea.