@treesareyellow - completely understand how daunting it all is, there are so many variables.
One of the biggest is the age of you/your partner. I’m going to answer as though you are in your twenties.
I agree with a pp. Priority for now would be saving the difference between your HA rent and a possible mortgage payment. I think you said the difference per month would be around £600. Even if you only save £4-500 of it for a year it would a) give you an idea of how it feels to live each month paying out that money; and b) would build up your savings for a deposit. Ideally, go for a savings account where you can’t touch it for that year, you’ll get a slightly better rate.
With regards to TTC, I can’t comment fully as I don’t have kids. But, from lots of posts on here and reading elsewhere, I would advise caution if you really want to buy a home. Having a child will play a factor in your affordability checks when applying for a mortgage.
I also agree with other pp’s re your first house. Don’t discount a 2-bed completely as your first home. When the time is right, it might be sensible to start off with a 2-bed. There is nothing wrong with young children sharing a room for the first few years of their lives.
The most important thing for now, given your HA rent amount, is to start upping your monthly savings - it is an advantage that you should absolutely maximise while you can.
And, when you do come to buy, be sensible. As long as the structure is sound, don’t worry about it being insta-worthy. A slightly crappy house in a good location is fine - take pride in making it your own over time. Structurally sound, neighbouring houses look ok, decent schools, feels safe - all good. Crappy decor, tired kitchen, unloved garden can all be sorted over time.
Also, when applying for your mortgage, check that lenders let you port your mortgage (ie staying with lender but moving to a new home). Can make things a bit easier if the time comes to move upwards from a 2-bed etc).
Am going to be very boring with my final comment, as it’s a repetition. Start saving the difference between HA rent and mortgage asap.