The only thing a divorce offers above a JS is the option to re-marry; starting divorce proceedings can also be an opportunity to re-visit the settlement terms of the JS, but it's rarely overturned unless it was pre-96 or the judge is satisfied that there wasn't full disclosure at the time of the JS. (Anecdotal evidence)
As the OP hasn't mentioned it, I'm assuming that he doesn't want to re-marry at this time; and as his former wife hasn't started proceedings its probably a safe bet that neither does his former wife at this time.
The OP hasn't said if he is paying spousal maintenance or not (I'm assuming not) and there's no guarantee that his former wife will ever re-marry in any event. So starting divorce proceedings on that basis is a bit airy-fairy really.
Assuming you are not paying spousal maintenance, then the money you are paying is a) child maintenance and b) your share of the mortgage.
Child maintenance you are obliged to pay anyway and since you got a variation on the basis of being made redundant, I'm assuming you had it reduced to the minimum level?
Your share of the mortgage payments must be quite small though? A mortgage of €25,000 is very low.
If you are paying spousal maintenance under court order - you are aware that you can claim tax relief on it? And you can also claim additional tax relief in respect of being a lone parent.
As I've said the only difference his living in the house could make is in respect to his assumed contribution to household bills and thus to the maintenance you pay and even then I suspect the difference would be very small.
Irksome as it may be, I don't think you will be able to overturn the property settlement of your JS on the basis of your former wife having a new partner.
If you want to be bought out, your best bet is to have a chat with your former wife & see if she would like to do so.