First I would remove the plant from its pot, closely inspect the rootball to be certain vine weevil is what you have. Flush the rootball under a stream of running water, removing all the old potting soil from the rootball along with the bugs and eggs. Repot in a new pot with new clean potting soil. If problem persists, you can use insecticides from the garden center.

vine weevil eggs are brown and less than a mm in size. so very difficult to see. the female lays eggs at the base of the plant next to the roots. as they hatch they have a ready supply of roots to feed on.
they are very unlikely to be in a compost heap unless you have thrown the plant & compost with grubs into the compost bins.

welcome to GoY too

if you have them in pots then follow the advice from Bathgate, the grubs can be fed to the birds/fish if you have a pond or squashed under foot. Keep an eye out for the adult weevils, they climb walls! squash them if you spot them.
Provado is a product that works.

If the eggs in your compost heap are white round and rather shiny and lots of them together they will likely be slug eggs.

As said, if you can see them, then they are not vine weevil eggs. In fact there are no soil pests in Britain which which lay eggs which can be seen with the naked eye.

I want to add you must treat this autumn with nematodes. Nothing is more effective, and it's completely human- and wildlife-safe, unlike many pesticides.
A £10 pack will treat dozens of pots. Vine weevils lay their eggs in late summer and prefer nice, soft potting compost to damp, hard soil - so plants in pots are really vulnerable.

Am I right in thinking the vine weevil can't fly? Would it help to grease round the tops of the pots? I get the wretched things in my patio pots and usually follow the procedure Bathgate suggests unless the plant is completely chewed off at the base before I noticed it was failing.(This happened to my favourite heuchera). Then I search through the compost and find as many grubs as I can before treating with Provado. Sometimes it's easier to ditch both plant and compost but I treat it first. Nematodes next autumn then!

Try putting a mulch of small gravel, grit, on top of the soil in the pots. It needs to be at least 2 inches deep. If the adult cannot reach soil then it does not lay eggs.
On single stem plants then you can cover the top of the pot with a circle of plastic (as they do for cabbages to stop cabbage root fly). The cover does need small holes to allow water to get in. Again, if the adult cannot reach soil near the stem then it does not lay eggs.

I am having a similar problem with these grubs. Reluctant to throw away good compost which has housed the grubs - there may be more undetected still there - l have immersed the compost in water for a week. One or two I have found after that treatment appear to be dead - no movement at all. I don't know whether unhatched eggs would suffer a similar fate, or whether they can survive in water.