This answer is rooted deeply in the principles of the specific Old School playstyle that the question is about, and which loosely define the OSR movement.
By making mapping a part of play, you can both increase engagement with the mapping, make the time spent mapping less separate and more overlapping with the “interesting” things, and decrease the absolute amount of time spent mapping.
But how do you make mapping a part of play?
The 1st correction: don't give mapping descriptions on sight
Mapping takes time. And I don't mean out of game; mapping takes time in-game in an OSR RPG. That means that proper (i.e., accurate, detailed) mapping can't be done just by glancing into a room. If you're giving foot-scale dimensions as soon as the party opens a door, stop. Give them a visual impression of the room, not a mathematical impression of the room, and keep play going.
Which leads to…
The 2nd correction: don't give away mapping information for free
Mapping is work. The mapper doesn't just glance around and start drawing a grid-perfect map. Accurately mapping a room or corridor involves pacing off the distance or getting out a knotted rope that's brought along just for the purpose. The party mapper measuring and mapping a room is as much legwork and moving around as the party expert/thief searching for traps or the party elf combing the stonework for secret doors.
Just like you don't give away the location and nature of traps and secrets just for the asking, don't give (accurate) mapping information just for the asking. When the mapper asks how long the corridor is, ask them if they start measuring it by slowly pacing down the corridor. At first you'll get a panicked reaction “No!” as they envision their doom in pit traps and on the blades/fangs of lurking beasties, but after a few repetitions of requiring the mapping character to actually acquire their desired map data in a real way, they'll start getting choosier about when and how they seek that data.
Maps and map data are a resource, and like every resource in and OSR game, the game kind of breaks when they're still required but made freely available.
You'll find that the mapper doesn't try to map rooms on sight anymore; they'll wait until a room is somewhat explored first. You'll find that mapping becomes the answer to “and what are you doing while those two spend three Turns checking the walls for secrets?” and begins to naturally overlap with such activity.
You'll also find that the mapper acquires the information more organically, making the process more part of the in-game activity itself than dry bookkeeping at the table. This in turn makes mapping information clearer, since the character is directly engaging with the dimension of the territory they want to record: if the mapper asks how wide the room is, ask them how they measure it; if they say they pace off the wall that the door they came in through is in, that's a mental image of activity and exploration happening, not just dry at-the-table bookkeeping. If they want to know how long the room is, they similarly have to choose the wall to pace off.
All that moving around the room is also in-game activity that can interact with the ever-present Unknown of a dungeon, which keeps the tension a bit higher than your average bookkeeping.
Does that sound tedious? Well that's actually a feature and self correcting, because…
The 3rd correction: don't assume mapping will happen
Mapping is something the players elect to do because it has utility — it is a resource they should be managing, not you. Don't assume or guarantee it for them!
Making mapping an in-game activity moves it into the realm of player responsibility. They want a map? Then they have to do the work to make the map. They don't like how long the mapper is taking to have their character strut around the room? Let them debate amongst themselves whether to make it stop or grudgingly put up with it for the greater good, instead of resenting you for the time it takes. Let the players choose to forego an accurate map!
Really, often they won't need an accurate map. The optimal behaviour is to map only as often and in as much detail as is necessary to be able to safely and quickly navigate around and back out of the location, and sometimes that means a boxes-and-lines map is good enough. How and what they map should be theirs to determine and devise. When you make that decision for them, all those times when the detail is unnecessary are, well, unnecessary, and turn the map into an imposition by you on them.
Let the players self-regulate mapping duties and game-time. When they really do want a good map, they'll invest their efforts into making one!
The challenge: refocusing on visual descriptions
The habit of initially describing rooms for the benefit of the mapper is a hard one to break. You might not be doing this to the extreme I'm about to depict, but the basic principle is the same regardless of degree: describe for the eyes of the adventurers and the mind's eyes of the players, not for the map of the mapper.
It's easy to say “it's a 20 by 20 room with a well in the north-east corner”, but that ease is like candy: it's tempting in direct proportion to its unhealthiness for the game. Save that kind of information structure for when the mapper has said “I spend a turn measuring and mapping the room while the fighter binds the wizard's wounds” or whatever. Toss those description habits and develop eye-view description habits:
Stepping up to the arch, your torches reveal a dead-end room. It's twice as wide as the corridor you stand in and square, with no other visible exits. In the far right corner is what looks like a low circular wall surrounding a pit or well.
The temptation is to info-dump, I know, but keep it simple to the point of actually missing details, and focus on suggesting a broad-brush impression of the room. This keeps it digestible and gives the players something to engage with and explore. It's a first visual impression for the adventurers, after all, so brevity is fitting anyway. As is usual in OSR games, the players will ask questions about features' details, which is akin to their character turning their attention onto the features within the game. (“What kind of wall is it?” “It's rough masonry, knee-high, and moss grows in the cracks.” “I walk over and hold my torch up to look into it…”)
And as more-or-less implied previously, if the mapper wants to map, using visually-focused descriptions makes acquiring mappable details just a different kind of question and detail that the player asks and their character inspects, instead of a separate category of table activity.
As a bonus, giving visual descriptions dovetails with the 1st, 2nd, and 3rd corrections: if the mapper or party wants a map faster, they can make a rough one based on the not-very-accurate idea of the space they get from the visual descriptions. There's no need to switch to Mapping Conversation Mode with you — they gather the details they want (if any) via exploration and your descriptions in response, seamlessly as part of normal exploration play.
Don't give the map details away at a glance, because you want the mapper to be acquiring this stuff through active interaction with the in-game environment, in a way that overlaps with and interacts with the other PCs' explorations and activity, and in a way that makes it the players' collective concern to manage and choose to do or forego instead of yours.
Give visual descriptions focused on what it would look like standing in that place, instead of descriptions tailored with a mapper in mind. Let the players drive the mapping, and let them do as much or as little as they all want. Let intra-party pressures determine the level of detail or lack thereof that they (collectively and emergently) decide is good enough for their maps.
P.S.: Don't forget the paper and ink
And it goes without saying that they can only map if they have the means and supplies. Don't give them maps for free in money or planning terms either! If they want a map, let them plan for it and buy paper, ink, quills, and ways of keeping them (and the resulting maps!) safe from unexpected tumbles into underground waterfalls.
Or not — again, it's up to the players now whether they want good maps or not.