Doulton or Martin Bros?

I think this is Doulton Lambeth, but there are no marks.

Maybe it's an apprentice piece, as it's unusual to see setting out marks (detail photo below).

It's a Tobacco jar missing a lid. They were made in large quantities by companies
like Denby and Langley, for retailers, and rarely marked, apart from the retailers
own name.
The glaze colours would be right for Doulton.

Thanks! I wondered what it was -- thought it was a brush pot.

I guess the TH is for a specific customer?

Yes, possibly a chain of tobacconists.
Larger ones were made for University collges, with an applied shield
which has the college crest on.

Its a tobacco pot missing its lid. Made in the late 1870's early 1880's I would suggest by Baileys of Fulham, the white beads are almost a trademark for them.
The initials are who it was made for, they made a lot of custom items like this and added names to order.
The reason is looks like Doulton is this was style of the day and Doulton was the biggest maker.
The reason it looks like Martin Bros is that RW Martin worked for them in the early 1870's. Nearly all early Martin brothers was fired at the Fulham Kilm and looks very similar. In fact the Martin Bros items marked Martinware are supposedly the range RW martin designed for Baileys. If your lucky some Baileys items have initials of the artist underneath but most are unmarked.
A quick google will show you some very similar. Still very collectable though and much more sought after than similar Doulton lidless items.