The key does lie in not sweating the details, but the trick is that which is the least intuitive one: positioning!
Follow three principles and theatre of the mind becomes much easier:
Use descriptive detail
When describing a fight scene, say in general terms where everything is relative to each other. You're not used to giving this detail verbally when your habits are tuned for battle mats, so it will take some practice and effort, but it pays off.
But most importantly, don't sweat being exact—exact details aren't necessary, and even entire things you miss or forget will get caught by the next two principles.
Be generous
When in doubt, rule in favour of the PCs. They're competent adventurers, right? If the Fighter was obviously moving into position last turn, assume that they didn't make any trivial mistakes about their positioning. Don't be miserly with the numbers either—if they're trying to get between the Wizard and the Orc and they've probably got enough movement to pull it off, then don't try to count exact feet to see whether they made it or fell short by 5 feet.
Don't sweat exact distances when the intent of the player is plausible to accomplish.
Track intent above all
What are people trying to do on their turn? Focus on that, rather than the bits and bobs of feet, ranges, and areas. Those are tools you can use as rough measures of whether an intent is plausible this round or will take more than one round. By focusing on what people are intending to accomplish with their allotment of time, it's easier to see what's plausible in the theatre of the mind, and you are actually imagining something more interesting. Is imagining 15 feet of movement an event worth spending imaginative brain power rendering in loving detail? Not really. But a careful repositioning to bolster the front rank, or diving away from a lumbering beast, or a steady advance toward the chieftain with deliberate steps bristling with murderous intent—those are worth time imagining.
So don't sweat the truly small stuff—feet and ranges and positioning in general and some kind of "mental grid". Those are just useful rules of thumb for making reasonable judgements about how people can move around, and trying to re-create the grid in your head is a waste of effort and defeats the purpose.
That's the short, pithy version for how to move from a grid-centric conception of combat to a functional theatre of mind that isn't just "now the grid is in my head and everything is harder, why would anyone do this?" For more nitty-gritty details on implementing a theatre of the mind when you've got a grid-based ruleset, my (and others') answers to these questions should provide:
Your PCs will likely not survive the Amber Temple fully intact. Even adding the two Winter Wolves is not likely to help, as they are only CR 3 creatures. That is, one Winter Wolf is a medium challenge for a party of 4-5 level 3 characters.
Compare that to the CR 12 monster in the first encounter, which alone would be a challenge for 4 level 12 player characters! Not only that, but the Arcanaloth is a 16th-level spellcaster, with finger of death and chain lightning both in it's repertoire, in addition to no fewer than 3 castings of fireball available to it. In 5 optimal rounds in this one encounter, you're looking at a lot of damage, certainly enough to kill at least one of your PC's.
Without doing the calculations (don't have my DMG at work...), two Winter Wolves are still only a hard (or possibly deadly) challenge for APL 3. You can't treat them as level 8 characters based on the hit dice, because level 8 player characters have classes which grant abilities at level 1, level 3, level 5, level 6, and so on, with level 5 being the most important as it's where the power level ramps up (fireball and Extra Attack come "online" at level 5). Winter Wolves don't have any of these advantages, and even with Pack Tactics and Cold Breath, it's unlikely to make a difference as the party is shelled by the highly deadly encounters within the Amber Temple.
The Amber Temple is one of two areas in the adventure with a recommended party level of 10, the other being Castle Ravenloft itself. See the Areas by Level table early on in the book (page 6). You can have a raven try to warn them away, similar to the encounter at the entrance to the windmill, but ultimately your players are the ones who get to choose.
That being said, if they are intent on entering the temple, let them. If they survive, so be it. If they start getting wrecked early on, you can hint to them that it might be wise to regroup and come back later when they're stronger, i.e., when they're more appropriately leveled and well-equipped. If they don't take your advice and push forward anyway, let them TPK.
Best Answer
Unique abilities need to be followed. In this situation, if everyone had the same movement speeds, etc., it would simply be straightforward. But, since you are in a situation that requires the extra effort to ensure that a player's movements need to be followed, to ensure they are not robbed of their special abilities, it simply means you need to spend that effort.
Going "grid-less" does not mean "map-less". When you describe your map to the party, there are several very important things to specify like distances and terrain; because they need this information in order to react to their environment. You can use a reference map - this does not need to be to scale, or particularly detailed, but it can help immensely for the players to understand their environment.
Tracking player "intent" is always going to be blurry. This is the same as declaring a "readied action" like "I am going to cast an eldritch blast at the next thing that walks through that door". What a player intends to do and what a player declares are two different things. In this situation, this is going to require both parties to communicate, in order to understand the layout, and the "intent" of the maneuver.
If you don't want to create a "disadvantage", keep everything further away. For example, if one player has a speed of 40ft, and the rest of the party has a speed of 30, make sure everything (at least starts) 40ft away. This means that the one player with the "extra maneuverability" can close the distance straight away, whereas the rest of the party needs to make extra effort (either by using their action to dash, or taking 2 turns) to close the distance.
But, is it really important? The most important thing to know is whether or not this really is an issue. Talk to the players and the GM and discuss whether or not this is important enough to always ensure this is kept track of 10ft of movement is one thing, ignoring difficult terrain is something much different.