Airline pushchair restrictions

[granolaelephant]

We've been given a lightweight pushchair (the Hauck sport - https://www.hauck.de/en/products/hauck-sport?slug=hauck-sport-blue-4007923171523 ) for an upcoming holiday, as our usual one is a very substantial all-terrain affair.
It seems fine for what we need it for, but the dimensions when folded are 81x25x45cm, as it's not an umbrella fold.
We're flying Easyjet there and BA on the way home. Not fussed about being able to collect the pushchair at the gate when we get off the flight (we're fans of the baby sling), but we don't want to have to pay extra for it as an additional piece of luggage, as that would defeat any money saved getting the pushchair second hand!
I've found the airline websites not super useful for this, with easyjet giving no info on max dimensions and BA just confirming it would be too big to collect at gate (but could we still drop it off at the gate when we board?).
Any experience out there with this? It's our first holiday with a baby so I'm feeling totally lost!

This is what easyjet says:

Flying with children and infants

We understand that flying with children can sometimes be a challenge, so we’ve tried to make it as easy as possible for you.

If you’re travelling with children under 5, we’ll board you early;

We can also carry two additional items for your child in the hold for free, if you require them

These include a travel cot, pushchair, car seat, collapsible or non-collapsible pram, booster seat and/or baby back carrier

(You usually keep your baby in the pushchair until you reach the gate and then they put it into the hold for you. It will get tagged when you drop your luggage off)

I've never known of anyone having to pay for a pushchair.

That's a great piece of information, I am a little worried Ryanair will moan if we turn up with our pushchair, as it separates into 2 pieces, ie, the folding frame base, and the sleeping/sitting up part.