Exactly what this would look like would depend a great deal on how (fast) the mass is accreted and whether there is a significant amount of energy accreted along with the mass.
There are two timescales at play here - the thermal timescale of the brown dwarf and the accretion timescale. If the latter is shorter than the former then the brown dwarf cannot adjust to additional mass and either inflates and becomes more luminous if the accretion is "hot" (if it accretes lots of energy too), or is compressed and becomes less luminous if the accretion is "cold" (most of the accreted energy is efficiently radiated). It will then gradually adjust its structure and luminosity on the thermal timescale of the new object. How the brown dwarf behaves is nothing to do with nuclear fusion here.
For slow accretion that nudges the brown dwarf over the limit then to see the effects of nuclear fusion you still have to wait for a thermal timescale - which is the timescale for the brown dwarf structure to adjust for a difference in energy input. The thermal timescale for an object at the substellar mass boundary is generally quite long, but depends on how old it is. That is because it is basically the gravitational potential energy divided by the luminosity. The former gets bigger as the brown dwarf gets older because it contracts (at a slowing rate). The latter gets smaller as the brown dwarf cools. A minimum timescale would be if the mass is accreted onto a young brown dwarf that is still strongly contracting. It will then approach the zero age main sequence on the thermal timescale of a relatively luminous $0.075M_\odot$ (star/brown dwarf) which is of order 500 million-1 billion years. What you would see is the brown dwarf/star get gradually less luminous over that time until it levelled off at its main sequence luminosity.
If you added the mass to an older, colder brown dwarf, then the timescale for adjustment would be considerably longer. The brown dwarf/star would gradually become more luminous over many billions of years until it reached the zero age main sequence luminosity appropriate for its new mass.