Have a look at the Australian Constitution
http://www.aph.gov.au/senate/general/constitution/
Note that the Australian Constitution is in fact an Act of the Imperial Parliament at Westminster. In particular read the preamble.
http://www.foundingdocs.gov.au/places/cth/cth1.htm
Now think of what has to be done to remove the monarch from our body politic and to replace the monarch with a President. Then think of our Parliament. We have a Constitutional Monarchy that is a representative democracy. We can have a republican form of government that is still a representative democracy.
We have an elected House of Representatives (lower house akin to the British House of Commons) but we have also an elected Senate akin to the US Senate. We have a mix of the two systems and that was deliberate. So our political representation wouldn't need to change much.
It would take a bit of work to replace the Monarch as the nominal head of the legal system but the model lies in the US for us so it shouldn't be too hard.