I highly doubt a problem with the duct would be causing a delay of the fan's operation. You should check the duct for any leaks which could be allowing the insulation from entering.

My suspicion dust or insulation in causing the fan blade to not turn freely. The fan motor itself may be failing. What type of switch controls the fan? Is it an on/off switch or a timer? It is is a timer, perhaps the timer is having problems.

The electrician can test the timer and the fan independently. When the fan is removed, see if you can vacuum the duct interior. If you see a lot of insulation you have a leaky duct. I recommend getting a Panasonic fan as a replacement. They are quiet and reliable.

You need a new fan. Tell electrician to step away from fan before he hurts himself. They only cost about 60 bucks at home depot.

The common fan moors are a simple shaded pole design, often with nothing resembling any type of actual bearings.

A pair of steel balls on the motor shaft and some sheet metal 'races' for them to rest against support the shaft.

If you are lucky they may have a dab of grease on the shaft supports.

Some are screwed into the fan housing, others are riveted.

Newer ones have a small cord and plug, making replacement easier (but I have seen more than one with a cord and plug riveted into the housing).

Buying a new unit and cannibalizing it for a new fan motor is often the easiest way to repair these things.
It eliminates having to remove the housing and not having the new fan match the existing hole in the ceiling.

25 feet is pretty far for theses small fans to drive air, even in smooth wall ducting (let alone corrugated duct).

"We have a crawl space rather than an attic, really, and the duct runs perhaps 25 feet across to a vent that is on the side of the house."

the bath fan is vented under the house??
I'm not understanding your post.

if you replace the fan panasonic makes a whisper
quiet fan. really quiet, nice product.
I wouldn't go back with a cheap loud fan.

best of luck.

A very belated reply -- I had responded but somehow must not have posted properly. To energy_rater, I should have said "tiny attic" -- it felt like a crawl space because it is so small! (I live in a cape cod.)

The fan is indeed bad, and I'm working on a replacement. I'd like a panasonic, but will go with the NuTone (what there is now) if it saves me doing a messy replacement and redoing drywall -- there is a big cutout there now.

Thank you all!

Hopefully your ductwork is in
good condition and properly sized.

The fan should be sized for the bathroom.

Here are some guidelines:

Small rooms:

For bathrooms up to 100 square feet in area,
HVI (Home Ventilation Institute) recommends
that an exhaust fan provide 1 CFM per square foot (approximately eight air changes per hour)
to properly ventilate the bathroom.

Example:

Bathroom is 8'x5' (with 8' ceilings).
Multiply 8 x 5 = 40ft.

Bathroom area is 40ft.
At 1 cfm per square foot the minimum
recommendation is a fan rated at 40 cfm.

Larger rooms:

For bathrooms above 100 square feet
in area, HVI recommends a ventilation
rate based on the number and type of
fixtures present, according to the following table:

Toilet 50 CFM
Shower 50 CFM
Bath Tub 50 CFM
Jetted Tub 100 CFM

Duration of ventilation:
HVI recommends that the fan be left on
for 20 minutes after use of the bathroom.

A timer is a good solution, allowing the fan
to turn off automatically at the proper time.

One thing you should know is most fans only
produce 50% of the rated cfm because of
undersized and poorly installed ducts.

Ducts should be hard pipe, not corrugated
plastic that will cause too much
restriction and possibly degrade and leak over time.

37 year veteran HVAC professional

Here is a link that might be useful: Heating, Ventilation and Air Conditioning Help and Advice

What a great question. My bathroom fan does the same thing. It worked well for the first 7 years we lived here and now it hesitates and then turns on. I am thinking mine is duct work and I'm going to rip it apart this weekend.

delays in starting up are not going
to be cased by duct work.

Take a shop vac and clean out the fan.

They can collect a surprising amount of dust that sticks to them when they are moving damp air.

Sounds like you need a new fan. But unless your house is very leaky, an exhaust fan's performance will always be limited by a lack of make-up air. Much more frequently so, in my experience, than problems with ducts. If you want to exhaust 50 cfm (cubic feet per minute, a measure of a volume of air) you need to allow in 50 cfm.

In good weather, crack a window in the bathroom or an adjoining room. When your HVAC is running and blowing in the bathroom, crack a window in a room with a return. For most houses other than those with horrendous air infiltration problems, it helps.

That's a fire waiting to happen, if it's not starting straight away - the motor will overheat. I suspect there's also a ton of dust in there too. Don't use the fan til it's resolved.

Why do you think its not starting right up. Mine does the same. I don't see any dust in mine though. I never thought about a fire though! That would be awful since the vent fan is in the ceiling.

Worn out bearings would be my guess, or an electrical issue - some electric motors require a "motor start" capacitor to get them turning, but I don't think your typical fan motor does. In extreme cases, dust buildup on the motor might restrict its turning. I replaced mine which was very noisy, when I realised how hot the motor was, I really worried about dust catching.

In short, old age.

Yes, a fire in a ceiling vent would be catastrophic. An apartment building near me was extensively damaged by fire due to dust buildup in a dryer vent....and bathroom fan ducts can attract lots of dust too.

The fire risk is one reason why rigid metal ducting is preferable for bathroom vent fans, as opposed to the plastic flex they use, it'll withstand/contain fire a little better, and the (relatively) smooth sides leave less dust traps. You're not supposed to put screws into dryer vents for that reason. I reluctantly did so with my kitchen vent I just installed, since most of what I read recommended it.

Bathroom fans are almost always shaded pole impedance protected motors.

Even if left at stall, they will not overheat enough to cause a fire.

The shaded pole design sets the direction of rotation and creates a very small starting torque without any capacitor.

I have not seen one with an actual bearing in a very long time.
Usually just steel balls on the shaft and some sheet metal they wear against.

Good to know, but the old motor from the fan I replaced (had to replace the whole unit, to get just a motor insert was the same price) was far too hot to touch, and that was with air passing over it. I think that's risky.

"To hot to touch" means about 130 F or higher.

Not high enough to be a hazard in most applications.

Start worrying when you can make a drop of water sizzle.