We are in a HA house and within a few months of moving in the HA wanted to do the checks for anything that needed replaced etc. (Windows, external doors, bathroom, downstairs toilet, kitchen) why they couldn't have done it in the few weeks the house was empty I have no idea. It came back that the kitchen and boiler were due for replacement, not because of the condition they were in but because of how old they were. (It did occur to me that they didn't need to see them to know how old they are, it was on their computer system, the contractors told us that! Looking at the things on the list we were surprised they said the boiler and kitchen as they were relatively serviceable, but the double glazing is in need of being sorted, there are gaps around some of the windows where you can see out between the frame and the window that sits in the frame, they need rehung (as does the back door) or replaced.)
Anyway, new boiler and kitchen. We were told that our HA replaces boilers when they are about 12 years old, regardless of condition or how efficient they are/well they work. The chap who replaced ours said it could very easily have worked for another 6 or more years, but the HA replace them at about 11 or 12 years old, so they wanted to do it even though it worked well. Even though yours still works your HA might be wanting to replace it because they want to do it at a certain age rather than do it when it conks out.
It was worth it though, we have a more economical, more environmentally friendly, easier to use boiler and now have a thermostat which we didn't have before. It only took half a day though, so didn't matter what time of year it was done. You can agree to have a new boiler but say it needs done at a warmer time of year, don't let anyone (HA or contractor) bully you into having it done in the cold weather if it is not suitable for you.
(When we had the kitchen done it took 2 weeks - full rewiring/electrics/lighting, plastering, all new pipes, tiling, flooring, painting walls, ceiling and doors, the works. We insisted it was done during school time, not holidays, as have 4 children and wouldn't be able to feed them during the day if they were at home for lunch. They made sure the cooker was rewired every evening and we could use it for cooking the evening meals, we were told we might be without washing machine and/or water for up to a week, but they would keep that to a minimum. I think we were without the washing machine for the Thursday and Friday, it was plumbed in soon after the pipes were replaced. The new sink couldn't be plumbed in until the cupboard and worktop were installed so were without that before we lost the washing machine but also had it back by the Friday and just had to take the dishes to wash in the bath as they didn't fit in the downstairs toilet sink. The contractors knew what we needed and tried their hardest to keep us cooking and washing while still replacing the kitchen. They couldn't promise to have things done by a certain day as if something went wrong it might take longer to get things done, but they did give us a rough time when we would have water, use of the kitchen etc and tried to stick to it.)
You can ask what boiler they plan on putting in so you can look it up online to see what it is like, but I would recommend you have it done, it should be better than the old one. Ours is noticeably better. Ask if you will be without heating for the time it will take to install, if so ask what they are going to provide you with (temporary heaters/oil filled radiators etc) to use as heating in the meantime. Get them to talk you through what will happen and what they will do to keep you warm.
As an aside, our HA services the boiler every 11 months. With our new boiler we thought it would be done 11 months after installation. No, the old one was due to be done a couple of weeks after being replaced so the new one was scheduled for the same time! We had it done, then within about 4 months they wanted to do it again. We said "no, it's been replaced then serviced recently." (I felt like the HA were constantly trying to demand access to our home, they'd been in a number of times in the 18 months we'd been here) HA threatened going to court for access and we will be required to pay costs etc. We said "ok, the courts will look at the dates and agree we are not legally required to have it done so soon." HA got the representative who dealt with our area involved, we told him it had been replaced and serviced recently and he agreed we didn't need to let them have access so soon and courts would agree with us. He did come back with (a possible story) that the HA were streamlining their system and wanting to have all the boilers in houses they own on our estate serviced at the same time so wanted to do it then, which is understandable. We eventually agreed, but made it clear we were sick of the HA wanting access to our home so they are only getting access in future for boiler servicing/gas check, emergencies and repairs we agree on.