What will happen on November 15th 2017, when the Segwit proposal expires? The time duration for the BIP-148 UASF is from August 1st to November 15th. During this time, the BIP-148 nodes will reject blocks that do not signal for Segwit.

Now, after Nov. 15th, even the BIP-148 nodes would start accepting all the blocks, whether they signal for Segwit or not. So, just after Nov. 15th, what would happen:

Case I: If the BIP-148 block chain is longer than the Legacy block chain?
(My understanding is, this won't even happen. Because if the BIP-148 network hash rate gets higher than that of the Legacy network, the Legacy block chain will always be "in sync" with the BIP-148 block chain, and the two block chains will be exactly the same. Am I correct?)

Case II: If the Legacy block chain is longer than the BIP-148 block chain?
(Now, after Nov 15th, the BIP-148 nodes start accepting all the blocks. The next solved block will most probably be a Legacy block. So, will the BIP-148 nodes switch to the "longer chain" [which is the Legacy chain], and discard all the blocks since the fork happened?)
Case I: If the BIP-148 block chain is longer than the Legacy block chain?
  (My understanding is, this won't even happen. Because if the BIP-148 network hash rate gets higher than that of the Legacy network, the Legacy block chain will always be "in sync" with the BIP-148 block chain, and the two block chains will be exactly the same. Am I correct?)


That is correct. In this case, there is only one blockchain and that is the BIP 148 chain. Since the BIP 148 chain is valid under the non BIP 148 rules, if it is longer, it will be the accepted blockchain.


  Case II: If the Legacy block chain is longer than the BIP-148 block chain?
  (Now, after Nov 15th, the BIP-148 nodes start accepting all the blocks. The next solved block will most probably be a Legacy block. So, will the BIP-148 nodes switch to the "longer chain" [which is the Legacy chain], and discard all the blocks since the fork happened?)


No. BIP 148 nodes will continue to follow their own chain and not switch to the legacy chain. This is because blocks between August 1st and November 15th in the legacy chain do not follow the BIP 148 consensus rules. Those rules specify that all blocks between August 1st and November 15th must signal for segwit.