You're writing formally, so the condition comes first: Should there be any delay, let me know.
It equals if there should be or if there is, with if dropped. In casual messages you'd say If there's a delay instead.
If you're writing to a customer or client
Writing a customer service email: when you need the condition stated first and your tone is formal, use Should there be any questions, please contact our support team. This equals If there are any questions..., with if dropped and should and there inverted.
Replying to a booking inquiry: use Should there be any delay with your order, we'll let you know right away. You're writing to a client who expects polished wording, so the inverted form reads as courteous and professional rather than stiff.
Drafting an official notice: Should there be a change to the schedule, the updated times will be posted. Here the inversion keeps the condition upfront in a register suited to announcements, which is exactly where this pattern belongs.
Texting a colleague or friend: stick with If there's a problem, just call me. The inverted form would sound overly formal in casual messages, so your reader's expected tone is the deciding factor.
A contraction rule for the inverted clause
In the inverted clause itself, avoid contractions. The main clause can contract normally. So in a customer email you write Should there be a problem with your booking, we haven't received payment confirmation. The haven't sits in the result clause; the inverted part stays uncontracted.
Notice the difference from the ordinary if-sentence: If there isn't a problem with your booking, we'll process it today. There the contraction isn't is fine. Once you drop if and front should, the contraction has to leave that clause: the clause itself must stay uncontracted, and that's the qualifier that decides your wording.
So when your condition is negative and formal, phrase it as Should there be no response by Friday, we'll cancel the reservation. Keeping no inside the inverted clause sidesteps the contraction question entirely, and it reads naturally in notices and client letters where the inverted form already fits.