Yes you can. Have a look at https://www.nationalrail.co.uk/on-the-train/train-travel-with-bicycles/cycle-policies/

So there’s no tool to book online a cycle space with the interrail pass… I will try to book it by phone then. 

There are a lot of train companies in the UK, it’s not so complicated in France 

Thanks for your answer! 

You can reserve seats and bike spaces for free where available and necessary, (see the above link for which companies/services need bike reservations) on www.gwr.com by creating an account.

I got it, thanks a lot ! 

You can book your bike spaces easily on LNER website for the East Coast line (ie Edinburgh or further north down to Kings Cross).

I find it easiest to do it on the desktop version of the website. You can just book the bike, no need to book a seat at the same time-however I suggest that you do. Your bike will be booked into the bike cupboards (4 bikes per train) in coach K. So book your seats in the half of K that is standard class or in the half of K that is 1st class depending on the Interrail pass that you have. Coach K is the only split standard/1st class coach on the train. You will automatically be allocated a seat elsewhere on the train but can amend this easily during booking. 
This absolutely works-have done it lots of times including this week between Newcastle and Kings Cross. 

Just scrolling and thought I can add to this, so apologies if you have already taken your trip.

You can't reserve bike spaces on Southeastern network (Dover to London), but there will normally be space on board. No cycle carriage, just space where passengers with accessibility needs normally go. If you’re using the Southeastern High Speed trains, look for the doorways highlighted with an orange stripe above.

I have to say, the variations in quality and quantity of train bike spaces in the UK is crazy. I see you're a bikepacker. Depending on the size of your bike, you may have trouble fitting in to some of the cupboards mentioned.
East Midlands I've generally been OK, and had the rear of a carriage that you can just wheel in to.
London North Eastern Railway (LNER) I have to book a double space in the cupboard as the tyre size is too big. To be honest, they're all made for road bikes or gravel. Anything larger you'll either need to deflated the tyre, or remove the wheel. The cupboards are tiny!
Some services may also make you remove your bags.
ScotRail you should generally have more space.

Hi! Thanks you for this reply even if I’ve already finished my trip indeed. I traveled from dover to Sittingbourne (train going to London Victoria) and then from Sittingbourne to London St Pancras. Each time there was a space where you can just lay the bikes on top of each other and secure it wit a strap. Was lucky it wasn’t too busy but the train can’t store more than 3 or 4 bikes this way. I read that it’s not allowed to bring a bike on the rush hours for the trains leaving or arriving London : https://www.southeasternrailway.co.uk/travel-information/on-board-the-train/cycles

I traveled on a LNER train from London King’s Cross to Edinburgh Waverley and I booked a space for my bike (for free). So I had to store it vertically in a cupboard outside the coach, which was locked by a controller. I have a big-sized bike so it was quite hard to store my bike like that, of course I had to remove the bags (like I do in most of trains). The other thing is I had to wait a few minutes at Waverley before I found someone to unlock the bike space, I wonder what would happened if I wasn’t leaving at the terminal station. 

Then I traveled with ScotRail from Edinburgh to Inverness and then to Thurso, lines where a bike reservation is required. The train leaving Edinburgh had several bike spaces : 2 bikes could be stored vertically with a hook and some would be strap (as in the LNER train). From Edinburgh to Thurso the bike was strapped outside the coach. 

Then I traveled from Mallaig to Glasgow with a booked space. In this train I had to hang my bike to a hook by his back wheel. Then from Glasgow to Ayr there’s no booking available, there’s hooks in one coach but there’re folding seats under. The train was busy and people were sat so I had to stand still with my bike just behind the doors. 

So I had no real issue to carry my bike during my trip to UK and trains were almost always on time… I think you can face troubles if carrying a bike with very large tyres. Mine is cross country bike but for this trip I equipped road tyres which are a little less large. 

I must say all the passengers i met were very nice and accomodating and I had a pleasant stay in England and Scotland! 

Good to hear it all worked well! Yes, you're right - you can't bring non-folding bikes on rush hour trains into London (morning) and out of London (evening). People still do on my line, even on really busy services.

I would certainly be worried about the cupboards being locked. I think it should be OK generally, because the staff should know what station the reservation is to.

Only problem I normally get aside from fitting the bike in is that when it's a busy service, the spaces are often taken by luggage, but staff are normally good about sorting it.