Hide glue is definitely the glue to use as, should the bridge need removing again at any future time, this facilitates the process.

You will need to carefully remove any residues of the previous glue from both the underside of the bridge and the top of the uke. If the previous glue was hide glue (which it should be from the maker), a heated, flat table knife will help with this. I would then slightly roughen the underneath of the bridge and, again carefully to avoid damage to the finish of the uke top, with abrasive paper of about 80 - 100 grade.

Once prepared, apply the hot hide glue (not too much) to the underside of the bridge and position on the top. You will need at least two clamps to hold either side of the bridge to the top while the glue sets. With a uke normal guitar clamps might be a little large to fir through the soundhole, so a little research to find suitable clamps might be necessary. Luthiers' clamps normally have a protective cork layer on the jaws to avoid damage to the finish of an instrument and, if you use some other type of clamp, e.g. one with bare metal jaws, you will need to fashion protective pads of some suitable material, e.g. cork or leather (or inside the instrument soft wood) to protect the bridge and uke from damage. Any residual hide glue can be removed using a dampened cloth before it sets.

Hope this helps a little.

Best regards,

mooman