Rules-as-written, you level up right away.
The PHB states:
As your character goes on adventures and overcomes challenges, he or she gains experience, represented by experience points. A character who reaches a specified experience point total advances in capability. This advancement is called gaining a level.
Note that it says "a character [...] advances in capability", not "a character can advance in capability". Contrasted with many other features that are optional, this means (RAW) that you can't delay your level-up.
Furthermore, Xanathar's Guide to Everything has a variant rule for shared campaigns, so-called checkpoint leveling (see XGtE p. 173). It's basically another alternative to XP, much like milestone leveling, so I won't go into details; its description says the following, however (emphasis mine):
At the end of a play session, characters must level up if they have reached enough checkpoints to do so.
This strongly suggests that leveling up as soon as you can is not only RAW, but also RAI (rules as intended).
I haven't found anything else targeting your issue in either the PHB, DMG or XGtE.
"Rules-as-fun", you probably won't run into any balancing issues as long as you talk with your DM and don't decide to level up mid-session.
Since the rest of your party probably doesn't want to wait, you'll be one level behind them. That's not a problem, it's quite common if the DM awards XP individually to everyone. However, you have to make sure to ask your DM if delaying your level-up is ok: it will affect their encounter balancing, especially if your DM normally makes sure that everyone levels up at the same time, for example by using milestone leveling.
For the same reason, mid-session-level-ups are also a no-go.
The DM may also have something planned that strongly requires you to level up. For instance, if you're level 4 and the big boss battle is coming up, you shouldn't be delaying your level-up, as level 4 to 5 is a pretty significant jump in power.
Therefore, if your DM says "no, you can't delay your level up", you should accept it without objecting—they likely have their reasons, not all of which they may be able to reveal to you.