Can I replace a fused UK plug with a regular plug?

The fuse (in the UK plug) is there to protect the cable (not the equipment - a common misunderstanding) so a fuse should be part of the wiring in whatever country of use. If it's not in the plug it will be in the equipment itself.
Clearly the Dremel does NOT have an internal fuse so simply taking the plug off and replacing it (with a fuse-less plug) is a dangerous solution.
Make up a small adapter - UK 13A socket (in a box) on a trailing lead that has the local version mains plug on it.
Not only will this keep the equipment 'safe' it will allow you to use other UK-purchased devices to be used should you get any more!

 

Fused plugs or otherwise not used in Aus.

 

Whatever local safety standards are in place are irrelevant to the thread and the OPs needs however the use of a fuse will mean the fuse blowing rather than the breaker on the board.

Fuses are put there for a reason, not decoration, but that's always down to what your local regulations call for and I'm not about to defend any particular countries practises over any other.....

What I described will work. Whether the OP wants to go to those lengths is for them to decide.

 

If the current rating of the mains cable supplying the equipment is less than the fuse rating that is fitted at the distribution panel then a separate fuse is recommended because other wise the cable could overloaded.
This is why in the UK we have fuses in plugs, because we have the ring main which is normally supplied with a 32 Amp fuse. Not 100% sure what they do in Europe and other countries but I doubt they use a 32 Amp fuse in the panel with no fuse in the plug.
Adam

 

Then perhaps armed with all the info available the op can make an educated decision.

 

Given the generic Dremels (here in the UK anyway) aren't protected by the fuse in the plug (that protects the cable, as already mentioned) then there will, of course, be some other 'fail safe' in the device somewhere - (here's a 'wild assumption') whether this is a thermal fuse (potentially) or a fusible link can only be guessed at subject to dismantling one - I may just do it!
I much prefer equipment that makes such 'fuses' available to the user without having to dismantle the item to access it internally..... and based my first (wild) assumption on the fact there is no fuse holder on the Dremels I have knowledge (and ownership) of!
I don't condemn countries for not using fused plugs neither do I support the UK's nanny-state inclusion of such protections - I was simply stating the facts as they exist.

 

The inclusion of a fuse for the UK market is because we have a ring main which has a 32 Amp fuse, that's the only reason I know. If we used a radial circuit in the UK with a 10 Amp MCB then we wouldn't need a fuse in the plug either. At the end of the day it's all about protecting the cable so it should be quite easy to descide what you need to do.

Thanks
Adam