Hi Kapil,
I have gone through a similar experience. A few things that could happen are as follows: if you do not serve your notice period, the company can label you as absconding or include negative remarks in your reference check. This could resurface during background verifications, requiring you to often explain to your new employer that it wasn't your fault and request them to honor your joining. If you are fortunate, you may secure the joining; if not, you might have to explore opportunities elsewhere.
Conversely, if you leave after serving a one-month notice, the company might provide you with a relieving letter and a service letter, possibly containing negative comments such as absconding or not completing the full notice period. You will have to continually clarify through your communication, like emails, that you did not act improperly.
The best approach is to persistently request a proper exit process from your current employer. Even if it means pleading for it, it is crucial for your future. Let go of any resentment towards the management. Additionally, try to keep your potential employer waiting for as long as possible; they can usually accommodate if you keep them informed. Leaving without serving the notice period should be considered as a last resort.
Best of luck!