When will we be able to get BT Sport and Eurosport/Premier Sports on Now TV?
If you want access to 'Eurosport' (as well as 'Sky Sports' via one of the 'NOW TV' passes) you can always pick up a Roku Express - currently £24 from Amazon UK, AO, and Tesco... or £24.99 from Argos, Media Streaming Market, and Very, if any of those are better for you - because there's Roku channels (i.e. apps) for both Amazon's "Prime Video" and "TVPlayer" through which you can watch 'Eurosport', although FWIW the one I prefer is Amazon's because it has 'Eurosport Player' with all the "Extra" options, whereas TVPlayer's app only gives access to 'Eurosport 1' and 'Eurosport 2' as part of their package. The only caveat with using "Prime Video" for 'Eurosport Player' is that you have to be an Amazon Prime member before you can subscribe to 'Eurosport Player' (at an additional cost) but it's been faultless for me so far and the picture quality is great - something that's not always the case when streaming content from 'TVPlayer', in my experience.
Unfortunately, there's no "Premier Sports" option on the Roku platform but, if you've got a 'Fire TV' device from Amazon you can always try watching via their 'Silk' browser app. If you go to the 'Premier Player' website (https://www.premierplayer.tv) in the 'Silk' browser and try playing the short intro/promo video that will tell you whether or not it should work before taking-out a paid sub and/or free trial. Basically, if the promo video plays okay, you should also be alright for the rest of their content when streamed via the 'Silk' browser on a 'Fire TV' device. You could also try the same test if you have a browser built-in to your TV and watch through that, if you have one, instead of an Amazon box or stick.
As to the matter of 'BT Sport'... from what I've read, it's going to be available through a separate Roku channel (app) which will be made available through the 'App Store' on NOW TV streaming devices - as opposed to streaming it from within the 'NOW TV' app itself, as you currently do when watching any 'Sky Sports' channels with a pass from 'NOW TV' - but AFAIK it's not likely to be made available until "mid-2019" so, by my reckoning that will be June/July time. The FA have set the start date for the 2019/20 PL Season as Saturday 10th August 2019, so I'd be very surprised if it wasn't available by then at the latest.
With it being a separate Roku channel (app) - released through the Roku-Powered™ app store on NOW TV devices - I'd expect to see an equivalent app also made available on Roku products in the UK at around the same time. That being the case, if you were to invest in a Roku Express as per my suggestion, you would be able to stream 'Sky Sports' (via the 'NOW TV' app), plus 'Eurosport' (via the 'Prime Video' or 'TVPlayer' apps), and 'BT Sport' (via it's own dedicated app) on Roku. The only one "missing" would then be 'Premier Sport', for which you'd need to use an alternative method of streaming - such as I have already described above.
Hopefully the question of streaming quality will also have been addressed by that time!