Hello, my name is***** and I have over 20 years of experience as a veterinarian. I am sorry to hear that Tiddles's base of his tail seems swollen and uncomfortable, but I am pleased to hear that it seems he is able to move it.
Does your fellow go outdoors unsupervised?
Or does he have indoor housemates that he plays or tussles with?
The most common causes of tail discomfort and swelling where a kitty can still move his tail but it hurts to do so are bite or scratch wounds to the base of the tail or tailhead area or anal gland infections or abscesses. These kitties have normal neurologic function, it just hurts to move the tail. I understand that you've tried to examine his back near the base of his tail, but he is too sore to let you do a thorough examination. Were you able to see if all of the swelling is on the tail itself, or is the area around his rectum swollen or abscessed (open wound near anus)?
The other possibility is that he is very constipated, and if the swelling is more around his rectum that may be hard stool stuck just inside the rectum. Has he been passing normal stools? Is he eating normally?
If he goes outdoors unsupervised another possible cause of his discomfort is a tail pull injury or tailhead trauma.
This occurs when a cat falls and lands on his tailhead/back of the pelvis or the tail gets caught (by another animal or a car) and pulled hard enough that the caudal spinal cord is stretched. Sometimes the nerves are just stretched, but in some cases they break and when that happens the damage can be permanent. Cats may not show the full extent of their injury for several days as secondary inflammation and nerve death occur. Treatment is with steroids and rest.
If he is having trouble passing urine or stool he should be seen by a veterinarian as soon as possible on an emergency basis. But if he is able to eliminate normally I recommend that he be closely confined and see his veterinarian tomorrow if he cannot be seen today.
They can examine him and prescribe antibiotics is a bite or scratch wound or anal gland abscess or infection is found, or they can prescribe anti-inflammatories if it appears this is a spinal cord injury.
Or if he is severely constipated (obstipated) they can sedate him and relieve that as well.
I do think that given how uncomfortable he is you simply won't be able to help him at home.
Best of luck with your fellow, please let me know if you have any further questions.