To interrupt without sounding rude, start with a phrase that signals you know you're breaking in. Excuse me gets someone's attention, and Sorry to interrupt politely admits you're entering their conversation. Say the phrase first, then your point.
Stepping into a conversation
Excuse me /ɪkˈskjuːz mi/ — Excuse me, Amina, I have one quick question.
Sorry to interrupt /ˈsɔːri tu ˌɪntɚˈrʌpt/ — Sorry to interrupt, but the bus is leaving in five minutes.
Not ""The bus is leaving in five minutes!" (cutting straight in)" — say ""Sorry to interrupt, but the bus is leaving in five minutes."". Lead with a softener like Sorry to interrupt so the interruption sounds polite, not blunt.
Try it: Lars: ...and then we can finalise the budget after... Yasmin: ___, but the projector is ready now. Which reply politely breaks in? (Sorry to interrupt / Well done / Never mind) Answer: Sorry to interrupt — 'Sorry to interrupt' politely acknowledges that she is breaking into their conversation before making her point. 'Well done' praises someone and does not fit here, and 'Never mind' means forget it, which is the opposite of raising a point.
Adding your point in a discussion
In a meeting or group chat, these phrases let you add an idea without cutting anyone off rudely. Can I just say softly introduces a comment, and If I could jump in asks for space to make a point. Both sound polite because they request the floor instead of grabbing it.
Can I just say /kæn aɪ dʒʌst seɪ/ — Can I just say that Mei's idea would save us time?
If I could jump in /ɪf aɪ kʊd dʒʌmp ɪn/ — If I could jump in, the budget has changed since last week.
Not ""Stop, the budget has changed." (grabbing the floor)" — say ""If I could jump in, the budget has changed."". If I could jump in asks for the floor politely instead of demanding it.
Try it: Mei: Alright, let's vote on the plan now. Diego: ___ one thing before we vote? Which reply gently adds a short comment? (Can I just say / Would you mind / Excuse me for) Answer: Can I just say — 'Can I just say ... one thing before we vote?' is the natural way to add a short comment to a discussion. 'Would you mind' asks someone else to do something, and 'Excuse me for' starts an apology (excuse me for being late), so neither introduces his point.