From your photo, the circuit board looks to be soldered to the jack socket and that would place the battery behind the bridge. Even if your hands were small enough, the battery would be difficult to change this far back.
Try this;
Unscrew the strap button from the jack socket, this should reveal the threaded barrel and retaining nut.
(You may need a bit of pressure to release the strap button, often they are secured using a thread lock liquid. You can achieve the same effect with a drop of nail varnish on the threads before re-attaching the strap button)
Carefully loosen and remove the retaining nut. you should mask off the surrounding area to prevent damage to the finish, also take care not to let the socket slide inside yet. Normally there are two small holes about halfway along the barrel, loop a fine cord or cable through the holes. this will allow you to draw the socket out again.
(Leave this cord long enough with a large knot at the end so it doesn't accidentally follow the socket inside the uke)
Let the socket drop inside and maneouvre it towards the soundhole.
Change the battery, draw the socket back and secure the nut and strap button.
Hope this helps
Vintage
Looks like you got a similar response from dwizum while I was composing this