Another twin mum here knowing how you feel!
This is how I did it - I would have a nosey first of all on rightmove for sold / for sale properties if there are any similar in your road / area that have had extensions and you can see how the space might work.
Then have a nosey at your local council's planning portal - you will usually find a "search applications" tab and if you put in your road name, or locality, you can often see the planning applications that others have made, and usually the plans are available too (for existing and proposed lay outs) so you can see what the council has / hasn't allowed.
I'd then get an idea of what you want (i.e. single storey / double storey) and work add roughly how much space you're adding. As a very rough guide, people use £1500/sq m plus VAT, you obviously have to double that if you're going to have a double storey extension. Thats usually a build cost, you have to add kitchen / bathroom etc on top of that.
That should give you a very rough idea of whether you can afford what you want. Then I'd see about approaching someone to prepare drawings for you (when you look at existing planning applications on line, you can see the details of the architect / company who has drawn up plans - its usually written somewhere on the plan so you can see if there is anyone local). Then its a question of submitting the plans for planning permission (or permitted development) and then finding a builder (usually better to get recommendations).
Good luck (and congratulations on your twins!).