See the NIH answer below on what's happening to grant budgets, at least for their agencies. It sounds like there is some budget trimming for the amount awarded for new grants, but they're trying to mitigate that somewhat. Where I think there's going to be a much greater impact is with new awards - with less money, and not wanting to hamstring grants with further cuts, they're simply likely to make less awards.
As for how much time before the possible effect of the sequester hits? It already has. Several people I've spoken to who do program planning, grants work, etc. for the Federal government have expressed the feeling that, because of the level of uncertainty about what money they'll have in the future, funding agencies are being very conservative about what they commit to spending. We could see this in the last budget cycle and the near shut-down - funding slowed to a crawl for a bit, and then when the continuing resolution got passed, there was a small "bump" as agencies spent out money they hadn't yet promised "just in case".
So if the sequester goes through, what that will really do is make those conservative, "We better not spend $$$ until we know we'll have it" plans a reality, followed by more severe paylines etc. in the next grant cycle.
So worst case: It's already here, we're just not committed to it yet.
Best case: The next funding cycle.
Edit: The actual answer has come from the NIH:
The NIH continues to operate under a Continuing Resolution as
described in NOT-OD-13-002, and therefore all non-competing
continuation awards are currently being funded at a level below that
indicated on the most recent Notice of Award (generally up to 90% of
the previously committed level). Final levels of FY 2013 funding may
be reduced by a sequestration. Despite the potential for reduced
funding, the NIH remains committed to our mission to seek fundamental
knowledge about the nature and behavior of living systems and the
application of that knowledge to enhance health, lengthen life, and
reduce the burdens of illness and disability.
Should a sequestration occur, NIH likely will reduce the final FY 2013
funding levels of non-competing continuation grants and expects to
make fewer competing awards to allow the agency to meet the available
budget allocation. Although each NIH Institute and Center (IC) will
assess allocations within their portfolio to maximize the scientific
impact, non-competing continuation awards that have already been made
may be restored above the current level as described in NOT-OD-13-002
but likely will not reach the full FY 2013 commitment level described
in the Notice of Award. Finally, in the event of a sequestration, NIH
ICs will announce their respective approaches to meeting the new