Overpay mortgage or save for new deposit?

[PeppermintInfusion]

I currently own a house with my mother (I didn't earn enough at the time to get a mortgage on my own) which DH and I live in and I/we pay the mortgage on (DM owns in name only).

Due to improved financial circumstances, now I could in theory get this mortgage on my own, I've paid down most of my outstanding debts, I've got some savings now, pension in place, end of student loan repayments in sight etc.
DH and I are now thinking of buying our own house within the next couple of years however as it stands I probably wouldn't get back what I paid for the house and we've done considerable work to the place.

Within the next few months I'll be in a position to start overpaying the mortgage. However, if we are planning to move (we would ideally need a bigger house in the medium-long term), would it be more financially sensible to put the extra money to overpaying the current mortgage or saving the additional towards our next house? I'm not sure which would be viewed better by lenders? Or is it just swings and roundabouts?

If possible I might try to keep the current house to rent until I could sell it, the going rate for rent in the area would be marginally more than the mortgage payment.
We do have a savings plan in place to have the estimated deposit (more than the minimum deposit) we'd need for our next house.

I do plan to go to an IFA about this down the line, but so far I haven't been able to find much on google about which would be preferred when taking a 2nd mortgage.

Not sure in your situation....but my view is to always reduce debt and so would pay down mortgage. We over paid on mortgage of last house which meant that we had a lot of equity when we sold, and that equity became our deposit. It meant we got a better interest rate, and so the further borrowings were a bit cheaper.

I would switch the current mortgage into your name only - i.e., get your Mum off the mortgage. Then pay down/overpay until you are mortgage free. Only then would I start saving for a deposit on a second property.

Keep your debt to a minimum - you simply never know when an illness, job loss, etc could put you into a financial mess so keeping debt to a minimum is wisest, imo.

It's partly a maths question. ie What percentage interest are you paying on the mortgage vs what percentage interest can you get for savings on deposit?

Earlybird, yes sorry meant to add that I will be probably switching this over to me only, which is why I mentioned I could get the mortgage myself in theory. If it came to it I would get it with DH, as we have always agreed that he would get out of it which he has put in to the renovations etc (Having a realistic discussion about worst case scenarios). We need to work out the implications for my DM as she has just retired. My main aim is not to take on too much debt, I'm not going to get a bigger house to be a slave to my mortgage payments.

Do you know if I changed it to my name only would I have to take out a completely new mortgage application or is it just a case of legally changing the name on the deeds etc? As it is a very low rate repayment mortgage that we probably only got because of my DM's status at the time. This has always been the plan with DM btw she only ever did it as a favour to me as she knew I wouldn't default.

Perhaps a good compromise might be to stay in the house a little longer (which I hadn't really considered before, so thanks! Though it might be a squeeze for a while with more children) and get the mortgage gone/really low as I'm sure it wouldn't count negatively if a lender could see I was being sensible with my money and putting it into overpayments? I could theoretically overpay even more by putting more of the money I'd be saving for a deposit towards the mortgage.

Cogito I think the interest rate is 4% on the mortgage, as I know it was 1-2% more than the best rate I could get on my savings which is why overpaying first became a consideration.