Any question answered...

There would be a no right turn sign if they couldn't, so my understanding is that there are no restrictions on traffic turning right into Bloomsbury Way.

The straight arrow on the traffic lights is informing road users that the road that is straight ahead is a one way street.

I think the one way signs only apply to what's in front of the traffic light, or in this case that's the intention: to stop motors turning left immediately.

Well, as with many oddly-shaped junctions in Central London, the signage and signalisation régime that was basically devised for more regularly-shaped streets doesn't quite work here. The problem is that there are two junctions within one here with streets into which a right turn would theoretically be possible, Bloomsbury Way (one-way eastbound) and New Oxford Street (east, one-way westbound). There's not enough space to define these two junctions separately, so the assumption is that once drivers have cleared the signage governing the signals at the junction of Museum Street and New Oxford Street (west westbound and east westbound), the junction of New Oxford Street (west eastbound), Museum Street, and Bloomsbury Way is a separate junction not subject to signal control. Put more simply, once you've cleared the traffic lights at Museum Street/New Oxford Street, you've also cleared the signage relating to it and you can go straight ahead into Museum Street or turn right into Bloomsbury Way. It is assumed that it is clear to drivers that they shouldn't turn into New Oxford Street (west eastbound)

There is no banned right turn into Bloomsbury Street, only into New Oxford Street (east of the junction). There is also a banned left turn for all modes except cyclists into New Oxford Street (west of the junction). Both of these banned turns only affect the 'first' junction.

The misleading impression that the right turn into Bloomsbury Way may be banned arises only because of the placement of the secondary signal head on the traffic island opposite (the one further back on the left in the pictures). Signalised junctions are required to usually have two 'primary' signal heads on each approach (in very exceptional cases one), which are usually placed either side of the entry of each street, as well as a 'secondary' signal head, which is often placed opposite the entry and can be quite far away. It still governs the same stop line as the primary heads. Its placement is confusing here, but there is no other place to put it.

The straight arrow on the traffic lights is informing road users that the road that is straight ahead is a one way street.

No, it means what mattioats says:

I think the one way signs only apply to what's in front of the traffic light, or in this case that's the intention: to stop motors turning left immediately.

(They're not a one-way signs, though, they just mean that straight ahead is mandatory, with the stated exception.)