What does “him as writ plays” (etc.) mean in old newspaper clipping?

An educated guess is that the quotation might be translated as:

[A man asked us] if it was Shakespeare - the one who wrote plays - that we were talking about, and if so, could he show us the very house he used, or at very least, all that was left of it.

In other words, the man speaking is offering to show off the place where presumably Shakespeare used to drink. Note that a "public house" is a bar, which would be called just a pub these days.

"The wery 'ouse" is a typical Cockney phrase. Dropped aitches were and are a feature of less educated English accents, and exchanging w's for v's was supposed to be the mark of the East Ender. (I've never spoken to anyone who's actually heard this, but Dickens, who was an acute observer, made Sam Weller pronounce his name Veller.)

I don't know what this book is, but it sounds [as an East Londoner] to be an author's attempt to reproduce a now archaic (From the Victorian era, up to the 1950's) uneducated [in grammar], East London or Cockney glottal-stop, dropped-'h' accent, I'll try to translate:

"who asked us: ‘if it was Shakespeare, him as writ plays, we was a torkin’ on ;"

"we was a torkin’ on" = are you talking about [this thing]?

"him as writ plays" = a man that writes plays, a Playwright [Shakespeare in this case] The speaker is trying to confirm they are speaking about the same person...

Becomes: who asked us: "Are you speaking about the playwright Shakespeare?"

Uneducated East Londoners tend to mix their tenses:

• "he who wrote becomes" "Him as writ"
• "we were" becomes "we was"

"The werry ’ouse he used, least ways, all as is left on it."

Is: "The Very (h)ouse he used [Public House - Pub, that Shakespeare used to drink in], at least, all that is left of it"

"After a twisting tramp through Cardinal Cap Ally,"

Is: After a [long] and winding walk [tramp, tramping the ground with your feet ('Plates of meat!)] through Cardinal Cap Ally - taking many turns down a labyrinth of back-allyways - we were brought out opposite the public-house [Pub] known by the name of the ‘ Smith Arms,’..."

This East London accent has softened if not disappeared following the widespread introduction of televisions in most homes and the influx of immigrants form around the world.

If not done well, this type of writing can seem a bit like Dick van Dyke's notorious english accent in the original 1960's Mary Poppins film, which tends to make most native Londoners ears bleed!