My condolences too Nic, that's pretty tough, it's difficult to know what else to say really,
but as you have given the start with the request for finishing advice, let's go with that. It's a very nice bit of Burr there, and an interesting shape too, as you know i very often make my pieces pretty glossy, which has a two fold advantage, done right, it stays shiny, this makes it easy to look after, and second, Joe public tends to like shiny stuff too, so it adds to it's appeal should you consider offering it for sale.
These days, especially with clean wood (i.e. not coloured with stains) I will pretty much always go with Shellac as a finish, it's pretty easy to apply and makes an excellent finish that looks top class.
I use Dewaxed blonde shellac flakes (or powder) and using a jar like a tiptree jam jar I put in about equal quantities of shellac to IDA (industrial meths) to a total of about half a jar, this makes a fairly robust solution (if it seems too viscous just add a bit more IDA till it looks good to you). With a piece like yours it might be best to make up a rubber, some rag or guncotton wrapped into a 6x6 inch square of lint free cloth (I tie mine off into a tight ball with a small cable tie) put a good handful of rag filler in, it holds more and needs recharging less. (With a plain bowl you can brush the shellac on which is much quicker, but it has to be cut back more)
A box of latex gloves are handy to keep the shellac off your fingers, it gets a bit sticky after a bit.
After 24 hours the shellac should be ready, just check it's all dissolved, (once mixed I put mine into an empty fairy liquid bottle, it makes it easy to control how much I put on the rubber) at first the rubber will just soak it right up, don't flood it, just keep adding it slowly till it has enough to let some out when you wipe it across the wood, you don't want it to leave more than a very thin layer, and with the burr, you want to avoid it pulling excess off into the little nooks and crannys in the form of drips, so be patient, keep the bowl in the lathe as it lets you have great control as you turn it slowly with one hand working your way round and wiping in a curved radial pattern from rim to base till you get back to the start. At first the wood will just suck it in, but as long as it is a bit dryer (usually it will be bone dry by the time you do a circuit for the first few goes anyway) just keep on going round, gradually building it up. If it catches in the holes and cracks have a bit of folded tissue handy to absorb the excess.
At first it will seem like it is never going to get done, but in fact it does build up surprisingly quickly, I can get the outside of a bowl done in about a half hour.
Once it looks well coated (all shiny but not too thick!) let it dry overnight, next day gently cut it back with some 400 grit just to remove the high spots till it all looks nicely matt, then go over it again with the rubber, 5 or 6 passes should be plenty, but judge it by appearance. leave again for 24 hours.
You then have two options to bring out the gloss, either use buffing wheels, the tripoli works well at around 1200, using shrt gentle passes and avoid lingering in any spot, if you get a build up of tripoli compound on the finish, don't try removing it with the wheel, sand it off with a bit of 400 and re-polish it with the wheel.
When it come to the diamond whit turn the speed down to 500 rpm, for some reason the diamond white will grab and instantly heat up and melt the shellac causing great pulls and tears in the finish if you use it at high speed. Guess how I found that out! They are most difficult to repair too.
So just work nice and slowly with the diamond white wheel, be patient, that's the secret.
You should need nothing beyond that, no wax or anything else, just an occasional wipe.
The alternative to the buffing wheels is burnishing cream, (Chestnuts sell a big bottle very reasonably, it lasts for ages) put some on a soft cloth and just hold it against the piece while spinning the lathe at around 900rpm, be check it often, it is surprising how much material this stuff can remove. The drawback with this is that the cream will get into the cracks etc, it is pretty soft when dry and can be brushed or picked out, but that is a bit fiddly, nevertheless it does get you a very nice high gloss quite quickly.
So, even if a high gloss doesn't appeal to you, at least you know how I get mine should you ever need to!
All the best Nic, hope this helps some.