Hi - I've broken it down into the three questions below. As ever with VAT, it's not a straight forward answer, as much can depend on the detail.
Does your website meet the new zero rating?
HMRC's guidance on the new zero rating is here: https://www.gov.uk/guidance/zero-rate-of-vat-for-electronic-publications. My understanding is that the intention is to extend zero rating to the digital equivalents of printed items that were already covered, for definitions see: https://www.gov.uk/guidance/zero-rating-books-and-printed-matter-for-vat-notice-70110. It's worth bearing in mind that there was a Tribunal ruling shortly prior to the new zero rating (News Corp) - in which the parties tried to draw a distinction between a digital version of a complete edition of a printed publication, and a rolling news service. With all this in mind, you're probably best placed to know how your website fits with these definitions.
What is the impact for overseas subscribers?
On the basis that your website subscriptions count as supplies of digital services, and the cross border sales are at least £8,818 per calendar year (this was a threshold introduced on 1 Jan 2019), the "place of supply" to your customers is where they are based. The VAT rate is determined by the rules in their country, not those applicable in the UK. The MOSS is just a "simplification" for reporting the VAT due, saving you having to register for VAT and submit VAT returns in every EU country you have customers in. Therefore the new zero rate in the UK will have no impact on your MOSS reporting.
That said, if your supplies outside the UK were less than the £8,818 threshold in each of 2018 and 2019 (and are likely to remain below in 2020), it's possible that you should have stopped MOSS from 01/01/19 and been including the sales in your "normal" UK VAT return. In which case the zero rating (if applicable to your subscriptions) would have an impact.
Did these new zero rating rules come in from 1st May 2020 due to Covid?
Yes. They were due to take effect later in the year anyway, but implementation was brought forward.
Apologies for the length of the reply, but hope it helps!