What can I do about a house with internal concrete floor with no DPM?
(I'm assuming that you are in the UK).

Concrete floors without DPM are not unusual in houses of that age, so there is a possibility that it's not a cowboy job. The broken bitumen you mention was probably the remnants of an asphalt DPM over the slab. The wood has (as you suggest) likely been used to level the concrete. If they were previously joists for a suspended floor, some have certainly been removed, as the spacing is way to wide for floor joists.

Without ripping the floor up, I can see two basic options:

1: Fill the gaps with concrete and get an asphalt company to lay an asphalt screed over it to restore the DPM.

2: Fill the gaps with concrete, lay a DPM over it, insulation, then a concrete / fibre screed. Note though that this will raise the floor level by 75-100mm (depending on the insulation).

In terms of concrete, as it looks like it's not a structural element, you could just use a "basic" 4:2:1 concrete mix (4 parts stone, 2 parts concreting sand, 1 part OPC). Add enough water to make it workable, but not so that it gets sloppy.