I realize this is kind of general as it would depend on the size of the white dwarf and the rate of accrual. The general idea I got thinking about is what would happen if a white dwarf star - lets say 0.8 solar mass, well below Chandrasekhar was to accrue a bunch of hydrogen rather quickly. I thought it might be get very hot and bright, perhaps similar to a red giant for a relatively brief period of time.
Mostly White dwarfs add mass by siphoning from the outer layer from a near by co-orbiting star, but I imagine, that mostly happens quite slowly, not fast enough to make the white dwarf shine like a main sequence star.
Are there studies on how quickly mass can be added in this way and how bright it makes the white dwarf? Does Hydrogen quickly fuse on the surface of a white dwarf or does it need to build up to a certain thickness first, perhaps creating a mini flash? or does the fusion happen rather quickly.
A very rough calculation, the escape velocity of a white dwarf, which would be similar to the terminal velocity of matter (mostly hydrogen) falling into it is about 5,500,000 meters/second for a 1 solar mass white dwarf. Source.
The velocity needed for hydrogen fusion is about 20,000,000 meters per second. Source (under the what makes fusion hard section). So, white dwarf's aren't massive enough to create fusion upon impact, but the high density and high heat of impact seems like it could create fusion relatively quickly as matter builds up around them, perhaps creating something that perhaps glows like a star and expands rather rapidly - assuming there's a sustained burn and not a kind of very quick explosion of sorts. I think it could glow very hot and perhaps expand rather large, but I'm just kind of guessing.
If a white dwarf was to crash into, say, a brown dwarf or very low mass star, hydrogen rich, say about .1 solar mass? Such a brown dwarf would be much larger than the white dwarf but also quite a bit lighter, probably light enough to get penetrated completely though the impact would also likely create fusion and it's difficult for me to imagine what would happen with so violent a crash. It seems to me that the density of the white dwarf, if it could retain most of the matter from the brown dwarf would be sufficient that you'd be left with a fast burning light weight fusion star, with a very dense core. This might be an impossible question, but how quickly would a .8 solar mass white dwarf consume the hydrogen from a .1 solar mass brown dwarf, should this rare kind of collision happen. Would it actually form a star-like object for a period of time, say a million years or so, or would the enormous energy of impact basically blow it apart no new star? Perhaps a near miss and an inside the Roche limit approach would be better than a direct impact for new star formation.