Having a 'Spent' conviction, custodial or otherwise, is not the same as a non-filtered conviction, by current UK law.
That is to say, your convictions of assault will likely be kept on an extensive list of filtered convictions, established by the home office, which includes ABH/GBH assaults, drug-trafficking offences and all sexual offenses. Assault convictions stay on this list for ten years after sentence.
Any employer in the UK, the commonwealth nations and the EU that performs a criminal background check of their candidates, such as in teaching for example, go through the UK DBS system.
If, say, you're going for a job as,a security guard or a "doorman", then the DBS in UK will provide a standard check where they only look at spent convictions, if any. However, if going for teaching or other work with minors in the UK (& commmonwealth, EU), then that's where the filtered/unfiltered list comes in.
Do note however that you can't apply for DBS, only your employer can. Good news there is, you don't have get that certificate authenticated.
If you need information of your whole record (cautions, convictions, etc), then you can apply for an ACRO certificate, which does have to be authenticated, if requested.
NOW, this is where it gets tricky. Most places outside of the EU and commonwealth nations do not have laws stating you have to supply a clean criminal record in the UK simply because they don't have treaty-access to UK police databases. It's entirely down to the FT candidate in vokunteering his authenticated ACRO police federation certificate. Otherwise, it's just a signed document promising that you have not committed any crimes in Taiwan or China (I expect it's the same in Vietnam, Cambodia, Thailand, Singapire. Not sure about Malaysia, SK).
In Taiwan (and in some instances China), it's down to the schools to request CRB checks, not visa offices, I think.
HESS school asks you of any convictions in their applications, but they're unlikely to check. It's very unlikely that buxiban schools will spend money or time, waiting weeks for the DBS certificate that they have to apply for.
There is a reform currently being pushed through by the ministry to tighten the CRB checks in light of the recent suicide of the author Lin Yi-han (林奕含). But, again, it will only require the foreigner to provide a promise of a clean record in Tawian, just I mentioned above.
If you're already in Taiwan, it's at your own discretion to reveal your convictions, not a legal requirement.
Not yet anyway.
EDIT (MUST READ): I've just called the DBS office, the UK government office that provides CRB certificates and they directed me to their sister-office in Scotland.
Basically, they said that they can provide a CRB check for international employer, but 1) only your employer can apply for one 2) they will only provide a Standard Check certificate which will not show any spent convictions. They will not do an Enhanced Check, which is meant for jobs-specific (i.e- teaching), only available in the UK and will show Spent Convictions that are on the Unfiltered Convictions list, such assaults and sexual offences.
The DBS also said that they do lisence to a number of outside agents to handle the DBS checks. Just google "UK visa agents intetnational."
But, they said, this process does take some weeks.
It's virtually impossible for any foreigner already outside of the UK to provide a CRB certificate in any format. Therefore, signing a document provided by your Taiwanese employer, saying you have not been convicted or currently investigated to your knowledge of a crime in Taiwan/China will be acceptable.
If they want to employ you, that's your only option.