Schengen visa refusal with an alert and chance of getting UK visa

If it's a Schengen Information System alert “for the purposes of refusing entry in the Schengen Area”, i.e. article 24 of Regulation (EC) No 1987/2006 (as seems likely), I believe the UK never had access to those. It did have access to other alerts in the SIS but currently doesn't. Appeals are strictly under national rules so you do need to follow whatever process Greece offers and, indeed, to hire a local lawyer.

However, you suffered two refusals less than two years ago, by two different countries, one of which was singled out as warranting an alert. Clearly, your circumstances were seen as risky. The UK doesn't have a reputation for leniency and your situation is unlikely to have changed enough to give you a good shot at obtaining a visa, quite apart from any alert. You need to evaluate that very carefully before any new application, otherwise you would just be wasting money and making matters worse.

In other words, the alert probably doesn't affect your chances very much but they seem very low to begin with. The Covid-19 pandemic doesn't help either, the British authorities will look at the purpose for your travel and will want to see that it is essential and absolutely cannot be postponed. Otherwise, it will look like a pretext and that will be another reason to refuse the visa.

Note that you will probably be asked to report prior refusals as part of the application procedure and trying to lie could also lead to very serious consequences (refusal of course but also a very long ban from the UK, typically much longer than bans from EU countries).