So as for nursery trade that goes back to the 1700s, when different eras and styles of gardens were developing, I wasn't able to find a great deal of information. The main reason is that many of the nurseries in Ireland have only been in business for the past 30 to 40 years. This seems to be the case in the UK as well. I then proceeded to go through a book titled the history of gardens, and it was entirely focused on the cultures, backgrounds, people and designs. There was nothing that had anything to do with plant production or usage. I decided to just look up the topic of nurseries in Wikipedia, and see what I could find. I got nowhere with that idea as well. 

So after some further searching I found an article that discussed tree production and forestry in the United States. It basically stated that before 1927, the year when the U.S. forest service was founded, there were no records of production, and any nurseries that did exist at the time were very small and had no means to mass produce various plants such as trees and shrubs. It was interesting in the fact that the wealthy, famous, and prominent tended to be very big proponents for growing and propagating trees, shrubs, and gardens in general. Two people that are perfect examples of this are both George Washington who planted tulip poplars on Mt. Vernon in 1740, and George Vanderbilt who grew a large number of tree plantations, many if not all completely from seed. 

Here is an image of the Biltmore Estate.

This is the "backyard" of the Biltmore Estate

Perspective of the amount of land that made up plantations

One of the trees that is part of the Biltmore plantations

This is a map of Mt. Vernon with the respective Tulip Poplars.

The same situation could be seen in Europe, where the elite and wealthy had access to plant materials, and nurseries would have been smaller and were probably devoted to the higher social classes. Running a nursery in Europe hundreds of years ago would have been a challenging task. There were no machines, the science of horticulture was no where as developed as it is now, and even just simple hand tools such as pruners would have been very crude if they had existed at all. The bottom line is that they would have been very labor intensive as they still can be today. In terms of propagation, seed was most likely used for the majority of production, and cuttings were probably somewhat common as well. As of now I can't completely confirm that information with any sources, but if I do find this to be true, I'll be sure to come back. 

Sources

Nix,  Steve. "Tree Planting in the United States - History of Wildland Forest Regeneration - U. S. Tree Planting Timeline." The Complete Forest, Trees and Forestry Home Page. N.p., n.d. Web. 4 Apr. 2012. <http://forestry.about.com/od/foresthistory1/a/tree_plt_timeln.htm>.