I would like to take some long-time exposure images from rivers and lakes, but I do not own a grey filter. That results in overexposed images, up to complete white images. Thus two questions: How can I still take those pictures without a grey filter, and how do I calculate the needed time for smooth water without having a grey filter?
You can merge multiple short exposure photos into a single long exposure image. There are a lot of tutorials on the net, for example:
Basically, he opens the sequence of photos in photoshop as layers in a single picture, then "auto-align layers", "convert to smart object" and "stack mode" - "mean".
Image alignment (the most time consuming task) can be avoided if the camera is perfectly stable between shots. If your photos are already aligned, basic stacking can be performed very quickly, even without sophisticated tools, for example this is how the free and open source ImageMagick does that:
convert photo1.jpg photo2.jpg photo3.jpg -evaluate-sequence mean result.jpg
You can limit the amount of light received by moving a black sock, piece of dark cardboard etc. in front of the lens. If you have a 10 second exposure and repeatedly cover the lens for a total of 8 seconds during this time, the net exposure will be 2 seconds, but the blur effect in the image will be something between a 2 second and a 10 second exposure because you didn't expose for 2 seconds continuously. This will obviously take some practice, but can be applied very creatively in different ways (e.g. to simulate a graduated ND filter by masking only parts of the image etc.)
Alternatively, you can also reduce light with transparent media other than an ND filter, such as black stockings or mesh.
How can I still take those pictures without a grey filter
One option is to try to DIY a neutral density filter. You won't get the best quality, of course, but you can get some useful ND-like effect by stretching a piece of Mylar film over your lens and securing it with a rubber band or two. Stretch it tight so that there are no wrinkles. If you need a darker filter, use two layers.
how do I calculate the needed time for smooth water without having a grey filter?
The longer the exposure, the more time you have for differences in the water surface to average out and the smoother it will appear. I'm a big fan of calculating things, but the empirical approach definitely wins here -- there are just too many variables and too much complexity to come up with a useful mathematical model.