You really need @AnkerOfficial help here. I recommend you contact Anker via email email@example.com because they are barely engaged in this forum. Takes longer for them to reply Here.
Any Powerhouse owners?
So I know about electrical principals and can cover better/worse.
It is 400+ Wh. It is 200W from the Anker reviews comments from Anker's Lisa. So you also need to be sure your demands are low enough.
You have 2 problems: charging the Powerhouse itself when in UK, and using the Powerhouse to charge UK mains devices.
I see a USA 3 prong socket, so any international multivolt PSU like on your typical laptop charger will work as it senses the USA 110V.
The page to buy says little.
It says "AC Output: USA 110V/EU 220V" which implies either a 220V specific EU model you must buy, or a switch to modify, but I don't see that switch.
On the Amazon site
I see "
AC output is 110V only, but voltage converters will work."
Therefore Anker's website appears to contradict Amazon site. I'd say probably Anker's site is wrong or there is a EU specific version you'd have to buy explicitly the EU version.
Your question is about UK mains adapter working. Well it depends. If your UK mains adapter is designed work in USA with a multisensing 110V/240V type written on the adapter then it will sense the Powerhouse output 110V and then work, and you just need to physically non-electrically plug adapter or use a US power cord. However, not everything in UK is multivolt, lots of items like say hair dryers are only 240V. If you use an external inverter to modify 110V to 240V then you are stacking inverters and power losses will cause even less Amps output.
Anker is not showing in support the download of the manual.
To charge the Powerhouse, I'm seeing as "DC Input: 16.8V/7.5A" and "Included Accessories: Micro USB Cable, AC Adapter". And that AC adapter specs are not stated, if it is multivolt sensing to work international then good, otherwise it won't work.
Based on the Amazon review, that customer got shipped a 230V sensing input AC adapter so that customer's Powerhouse can be charged international.
Also here they say "The input of the adapter is 100V~240V"
Based on the small amount of information Anker has made public, I'd say you cannot be assured of it working for you in UK.
Seeing as its not sold in the UK you'd struggle to get support, warranty. If you found the device worked but not for your device, so Anker product not defective but you're not happy, then you can't ask for refund.
If you import it then you may well have to pay duty, VAT, fee. Given you may struggle to get support, it is an expensive risk.