I think the whole area is gone.

I will look further, but I
think the area where Alfred Place used to might be around where PC World and Somer Community Housing Trust Buildings are today. All new builds I'm afraid.

The enumeration district was bounded by Lower Bristol Road to the south (still there today), the River Avon to the north (still there today), Mill Lane to the west. (I'm not yet sure exactly where that was but there is still a Twerton Mill which may have been on Mill Lane?), and Dredgers Bridge Road to the east (hard to say where that was - perhaps what is now Fieldings Road which leads to a bridge crossing the river?).
There are very few roads mentioned in the enumeration district - Longmead St is one which appears to be quite long but I haven't located it as yet. There are lots of 'Buildings', Villas and houses named but nothing that can be used as a landmark to identify where Alfred Place used to stand. It was obviously on a corner though as you'd expect of a pub numbered "1".
I have to go out now, but will do some more digging later.