The poster talks of ‘a skeleton key’: does that actually mean a skeleton key [for a warded lock] or does it mean a bit key? By ‘barrel’, does the poster mean a pipe key?
Bramah locks are distinctive and easily recognised from the keyhole.
If a key is available which turns in the other locks, another possibility exists.
Robert Barron also invented, but did not patent, a ‘fly-talon’ lock, soon after the double-acting tumbler invention, so probably in the late 18C. He made these in London, with wards, so a skeleton key might operate. After the bolt is thrown, an additional turn of the key moves the fly talon. Then even the real key, or anything else which will turn in the lock, in the unlocking direction, will turn ineffectually. The spring pressure on the locking tumbler or levers will be felt, but the key will not move the bolt. In principle, this ‘fly talon’ lock was re-invented by Gould in 1817. Gould locks were made with both a single tumbler, and also with 2 or 3 levers.
Gould’s version is described in PRICE Treatise on fire and thiefproof depositories pp372-373. The fly-talon is a copy of Barron’s.
To unlock, the key is turned twice in the locking direction, then once in the unlocking direction.
The counter-intuitive operation probably rendered such locks unpopular. However, Barron certainly supplied some of his for late 18C London-made furniture. Barron locks, incidentally, often have hook wards, and sometimes interlaced hook wards so that they cannot be skeletoned.
If a bit key is available which turns in the other drwers, try turning it first in the locking direction. On ordinary tumbler/lever locks, it will not turn! If it does, either the bolt step is too short for the lock, or this is a fly talon lock. The fly talon has a spring on it so it operates on the ‘over-centre principle’, (and this spring can break). Note that on a Barron tumbler lock, the bolt step is the step farthest from the backplate, unlike other lever cabinet locks in which the bolt step is the one nearest the backplate.
One point bothers me though. Late 18C - early 19C furniture of good quality was unlikely still to be made of oak. The wood of choice was then mahogony. Late Victorian/Edwardian repro oak furniture would be unlikely to have a fly talon lock. Also, bureau carcases are rarely so strong and stiff they cannot be spread apart by sash cramps (possibly with the aid of wedges initially) … . The throw of the bolt is quite short. And if the item is oak, no variety of Barron lock is likely to have been used. (That said though, anything is possible.)
All things considered though — a Bramah lock on the top drawer seems most likely.