I have had grass cutting issues as a tenant myself but I had a landmower supplied with the house as part of the inventry when I signed the agreement. So when the landmower broke it was his responsiblity to fix the mower or buy a new one, which he did but after about 1 year delay. In that year there was no way of borrowing a mower and the landlords agent said one was on its way every week so we thought nothing of the delay. I grass grew really long and it was too long for any mower so a professional gardener came in and removed 2 tons of grass and it cost hundreds of pounds all at the landlords cost. The delay was down to the letting agents not doing their job and the poor landlord paid.
I found out alot of other tenants with the same letting agents had the same broken lawnmower issue which again went unsolved until it cost the landlords hundreds to fix.
I think the only thing to suggest is get a really cheap mower on offer and share the cost with the other residents. Otherwise if it over grows so it cannot be cut with a mower you might have to pay for the gardener, cheaper to buy a cheap mower and as old tenants move out they get maybe some money back and the new tenants pay alittle towards it when they move in therefore you have a movement of money round and you only pay a little towards the mower if you still live there.