You might need to look for other cities in case you can't find a next day appointment where you are.
And just to add to your earlier question, although you can apply for ILR Long residence later, you must be in possession of a valid visa at all times and at the time of application. In short, your visa must not expire before you apply for ILR otherwise, you risk losing all ten years. And as you have been told, you cannot apply for ILR outside UK and you cannot travel until you have received your ILR approval. if you travel while your ILR application is been consider, the application will be nullified.
Your option is to apply for ILR with Super Priority and then try to get UKVCAS appointment quickly. You can consider Glasgow, I remember they used to have slots (pre covid). Otherwise, look at other UKVCAS centres and be ready to travel to the next available centre.
All the best