The whole orientation thing is over egged. As you correctly intuit, the sun you receive has more to do with obstacles than orientation. In fact, the only relevance of orientation is that you may have to take the house itself into account as an obstacle. What you really need is to be able to calculate the length of a shadow, including that of the house.
Here’s a particular house with a directly north facing back garden. The pitch of the roof is important. Once the sun elevation is greater than the roof pitch (38 degrees in this case, which is typical for an Irish roof) then the obstacle you are measuring is the gutter line, not the roof line.
So in this case a 17’ gutter line casts a 22’ shadow at midday on April 1st, a 16’ shadow on May 15th, and a 10ft shadow at mid summer. In general, for an object of height x, the shadow length will be x / tan(α), where α is the solar elevation angle.
You can get the noonday solar elevation for any location and day of the year from timeanddate.com. Of course, it’s more complicated than that. The sun elevation varies throughout the day, as well as the year. The noonday elevation is a maximum. You would need more complicated tools to show the shadows for a particular time of day.
The web app that spacebaby posted will give you the solar azimuth (direction of the sun) for any time of day and date. That will help you identify which obstacles are the important ones, depending on how/when you plan to use a garden. As you say, trees, walls, and other obstacles are just as important as the house itself. An architect would have shadow analysis software for figuring this out.