Flybe probably has an an interline agreement with the 1st airline. See
Flybe | Corporate | Media | News archive. It is possible that 1st airline will not send luggage on with two tickets. You will find this out when you check in at BNE, but as others have recommended have all of your details readily available.
This used to be a no-brainier, if you could show tickets with an OK status they would check the bags through. More recently some (larger) airlines have been refusing claims when bags go missing as 'they don't have a single contract' meaning that some (smaller) airlines have had to absorb significant Warsaw/Montreal claims when things go pear-shaped.
As an example BE would be responsible to you to pay your claim if bags lost/damaged even if 1st airline mishandled at SIN. Normally BE would claim from 1st airline but if they refused they would be stuck for the entire amount. Because of this some airlines no longer accept through checking.
Wasn't it much better with binding IATA Resolutions, before the ACCC, the EC and the US DOT took those away? (Rhetorical question, no need to respond)