Could you machine tie? There are different ways to do this. One is to place a piece of yarn over the spot, then use a machine zigzag in place to secure it. If you use a regular zigzag, then you need to do about 4 very tiny straight stitches at the beginning and end to secure the thread ends (so they don't pull out). Or, you can choose a decorative stitch that incorporates small securing stitches at the end (some machines have stitches like this).
I would not try to hand tie a t-shirt quilt; too hard! Usually t-shirt quilts are made with stabilizer behind the shirts, and this makes pushing any kind of needle through by hand problematic. Even if your sewing machine is small, you should be able to maneuver the quilt into position for machine ties.
Also, you do not need to cut the thread between the ties. You can move from one to the next in an organized manner and cut the threads after you are done with a section.
After the yarn (or floss, or whatever you decide to use) is secured to the quilt by machine, you can tie it so it is still hand-tied.