What is the best method to graft finger limes?. The branches are so thin. I tried a couple of bud grafts and the area around seems healed but the actual bud is not green but taking on the characteristics of the root stock. The buds are not loose but fused to the root stock. Not sure if they are dead or alive. These were unwrapped about 12 hours ago, after 3 weeks. Not the best work I have done but I'm very new to citrus. Any recommendations?

They are difficult, but I T-bud them without a problem.

Budding can be first done when the stem is 12 mm in diameter, and at 10 cm above the soil line. - Millet

As long as there is some green at the bud node then there is a chance they will grow.
It looks (not quite certain) that you may have taken a node with a small shoot already growing, rather than just a leaf node. If so, what can happen is that as the shoot dies back, the bud at the node also dries out and dies. Then it will never grow.
It is possible to T-bud microcitrus, but as Laaz says it is difficult. I find a 'shoot graft' (not sure of correct name) works better. I take a two or three inch twig of the finger lime and at the bottom make a sloping cut right through about an inch long. Same thickness rootstock, remove top completely, again make sloping cut an inch long. Bind the two cut pieces tightly together, wrap whole area in tape or plastic bag and leave in shady spot about a month. They should unite together and new shoots start to grow.

From your photos, it was a bit hard to see the actual bud eye(in a bright spot). The buds seemed healed in ok, but unless the bud eye starts to swell, they may sit there for ages. Usually if the eye has died, they will rot or scar, so should still be ok, unless they really dried off after tape was removed.
Fingerlime buds can sit till next season till the buds activate.
For very small buds I prefer Buddy Tape, a self adhesive tape, which doesn't need to be removed for bud to sprout, also doesn't have to be tied tight which may damage the bud eye.
Also I found chip budding easier to do with small buds.
You may try to bend the top of the rootstock over, above your bud to force its growth, rather than cut the top off.
Another method to force buds is to cut a notch about 1cm above your bud.