If you decide you do still want to wean, then :
Yes, at 12 weeks the lambs are perfectly capable of getting all their nutrition ruminantly. Up to you whether you give them some cake, or grass pellets, to make up for the milk they will be losing. As long as you have good grass for them, they will be fine.
If the ewes and lambs are in the next field to each other you are likely to get heads stuck in fences, lambs ramming fences, etc, so it's best to have at least two fences (ie., one clear field or other ground) between them. If you can put them out of sight, smell and as far as possible sound of each other, they'll adjust quicker, in my experience. If they can hear each other call, the calling goes on for a couple of days, sometimes a bit longer.
It makes it easier on the lambs if you do these things :
* leave them where they were and take the ewes away, so the lambs know the ground, where to shelter, etc. Just for a few days till they stop crying and settle into being grazing animals again, then move them onto the fresh ground you have ready for them. They won't eat a great deal of grass for a day or two while they shout their heads off anyway
* have a couple of older sheep with them to be "aunties". Maybe ewes without lambs, or last year's lambs, or a couple of ewes with lambs who don't need to be speaned because they have plenty of condition and or you are not going to breed them this year. (Assuming you don't have an "unringed tup lambs approaching sexually mature" problem).
Yes the ewes will need to be "dried off", so put them on slim pickings for a couple of weeks. Some people put them in a yard or shed with straw for a few days, then, if the grass was getting short where they were, you can put them back where you took them from after you move the lambs to better ground. Grass makes milk, and cake makes milk, and you don't want milk; hay and straw don't make much milk, so if you think they need a bit more food than the grass gives them, give them some hay or (better) nice barley straw to top up with.
"Gimmer" is the sheep equivalent of "heifer". It's not an age, it's a condition, basically a maiden female. So you can have a gimmer lamb (ewe lamb), gimmer hogg (ewe hogg : after her first Christmas and before her first shearing), gimmer shearling (after her first shearing), etc. As with heifers, they are confusingly often still referred to as gimmers (or heifers) when they have their first lambs / calves
So the advice you have been given relates to weaning first time mums earlier to give them more time to regain condition before going to the tup for the second time. It's definitely something I would do if I lambed hoggs (tupped them as lambs, so they had their own lambs when they were just 1 year old), and with first-timers who reared twins and have put a lot into their lambs, so look a bit scruffy and scrawny.
The rule of thumb is that it takes a fortnight to see a change in BCS. If you are weaning them, they will be on slim pickings for two weeks, but not producing milk, so their condition will probably stay about the same or improve very slightly for the first two weeks after weaning. Then check them, and anyone who seems skinny might want better grass, a bit of cake too perhaps if the grass isn't great. But you have quite a long time yet before tupping, so you don't really need to be in a hurry to put condition on them unless they're
really poor. If you will want to flush them (to get as many multiples as possible), you need a bit of leeway to have them on "a rising plane of nutrition" (ie., putting weight on) as you put the tups in, so don't overdo the feeding up now
ETA, if it takes a fortnight to add 0.5 to BCS, and it's 4 months to tupping now, and you want them at 2 or 2.5 for tupping, you can see that you really don't need to be in a rush. Although of course they will add condition quicker on summer grass and you might need to use a bit of cake to add condition come October.