I am alittle worried about my Chinese Elm...The tree seems very happy where it is at the moment, but my problem is, I have just looked at it and its covered in new leaves and shoots,
Sheesh, some people are hard to please.
Just kidding, all too often we see trees that are hanging on by a thread, if yours is actively growing count your blessings. No need to rush into anything, remember it's a tree and operates on a different time scale than you might wish. Think in terms of weeks and months for even minor changes to become apparent and years for substantial changes to show.
The two best times to re-pot a Chinese Elm is the spring or early winter. Let me explain, most deciduous trees (which Chinese Elms can be) are re-potted just prior to leafing out in the spring. This is when I re-pot my Chinese Elms.
Since this species is a sub-tropical (some might say closer to temperate) it can be handled differently if you choose to do so. An alternative strategy is to grow it outside all summer and fall, allowing the tree to enter dormancy. After allowing a brief dormancy the tree can
be brought indoors for an early spring. This would be the other appropriate time to re-pot, again just prior to it leafing out. I put the word 'can' in Italics in order to alert you that this may not be the best course of action. If your heart is set on having it inside you can with proper lighting and other appropriate care. But, the tree will be just as happy to spend the entire winter dormant.
After having written all that I will say that I have re-potted this species while in leaf with good results. You should weigh the potential benefits against the risks involved. Exactly how root-bound is the tree? How bad is the soil, and how small is the actual pot?
My reasons for wanting to do something about the roots is because, the first time I watered it, I wondered how deep the inside pot was, so I went to lift it out, and the plant came out of the inside pot completely, with no mess, it didn't look too pot bound, but, it was very very dry.
A tree that can be lifted out of it's pot with an intact rootball is not necessarily a problem. The fact that it was dry falls on you. You can manage a tree in a less than ideal soil, it comes down to proper watering. The old adage about watering multiple times becomes more important the denser the medium is.
A modern, free draining medium can be watered much easier than a finely textured, denser one. If your tree is in a relatively dense medium, which it very likely is, then watering twice or even three times in quick succession will help to ensure that all of the soil is thoroughly saturated each time you water. This should always be your goal, no half measures when it comes to watering, it's all or nothing.
There were quite a lot of roots, and I think the tree looks as though it is in too small a pot?
A good intact rootball is OK, a badly root bound one starts to become a problem but it is a gradual thing. A 1 1/2" trunk in a 4 3/4" pot does seem rather small but we don't know how long it was in that pot. Your tree was likely field grown and only recently potted in order to be sold. Sometimes a smaller pot is better in that it allows the medium to dry quicker. The ideal pot size/medium type combination is one that dries, and therefore requires watering, daily. Not saying you should water every day just that a large mass of soil that stays wet for extended periods is not appropriate. Often large trees that are collected are potted in the smallest pot that will accommodate the rootball.
I'm not trying to discourage you from re-potting just to make you aware so that you can make an informed decision. If you are leaning toward re-potting you will need to locate a pot that is appropriately sized and appeals to you. You will also need to acquire (either purchase or mix your own which is a whole other kettle of fish) and determine when you are going to do it. Until you have the appropriate supplies on hand, the point is moot. Have you read the thread about what constitutes a good bonsai medium?