"Old" router

The bandwidth required by Zwift is negligible so I wouldn't bother changing for that reason alone. 

If you are generally finding that your WiFi at home is poor because of coverage or too much demand from users (kids streaming generic pop warbling videos) then upgrading would help if you purchased a router that operated both 2.4GHz and 5GHz (we have our phones and tablets on 5GHz at home and the rest on 2.4GHz). Also if you can get a router that uses the 802.11ac standard then you can theoretically get Gigabit+ speeds although in reality you won't.

However, Wireless AC is really only useful if you are streaming/moving content between devices on the network, the limiter to download from the web will be your ISP speed which in most areas of the country is d*gsh*t.

Buy a new router for new features and security, but your zwifitng won't be affected that much if your broadband speed is poor.

Check the speed you're getting with an app.
If it's less than you expected then get on the phone to your provider. They often send you a new server to replace the old.

2 things, some ISPs only let your use their routers and you need to check if you need a cable router or not.
You could consider a wireless access point if you are streaming to a tv via a stick or box or even better cat6 cable.

I've had fairly negative experiences with ISP supplied routers - especially TalkTalk's Huawei heap of junk, which was only able to support one device at once.

I spent about £70 on a new router, and the speeds I was getting under normal use rocketed

The one I got was: TP-Link AC1200 Wireless Dual Band Gigabit VDSL/ADSL Modem Router for Phone Line Connections on Amazon

This was 6 or so months ago now, so there's probably something newer and more exciting, but this seems to do the job for me.

Set up was simple - plug the cables in, pick TalkTalk in the ISP dropdown, and pick a WiFi name and password.

Your mileage may vary depending on ISP, as well as the size of your house and location of your router. The BT homehub tends to get good reviews. If you live in a big house, or you are zwifting far away from your router, you may want to buy a range extender, or one of these schnazzy mesh network systems like google wifi, though I haven't used these, so I can't comment as to how good they are.

I'd guess that it would.

A 5 year old router might not support 5GHz (depends on which model/supplier) and in general 5GHz will give you faster speeds though for a shorter range (i.e. doesn't cover as wide an area as 2.4GHz or go through brick walls as well).

For most applications, your router won't be the bottleneck unless it's old/sub-standard etc. Most of the time it'll be the speed of your broadband or most likely the contention ratio. For those who don't know, contention ratio is how many times your ISP can "oversell" the internet access with the assumption that not everyone wants peak speeds at the same time. Typically, you can't change the contention ratio without changing ISP or switching to a business service (more expensive).

Personally, I'd recommend switching your ISP as that will give you a brand new router and most likely a better deal for internet access.

Probably*

It will depend on what type of wireless network your current router supports.  As a rule of thumb, speeds get faster and range gets greater (and less subject to inteference) as time goes on.

Note that not all receivers are capable of the latest and greatest - for example a old laptop or phone won't necessarily support the same wireless network types, so may end up having to downgrade the signal to something slower.

Replacing the router is not the first thing I'd do I have to say - for the most part it's much easier to let your ISP supply and configure the router unless you've the skillz to do this yourself.  If they've not updated it in 5+ years then I'd phone them up and ask for an updated router - pleading poor performance etc.

Sky replaced a three year old router for me with no quibbles/cost whatsoever as an example - and we can now happily stream HD upstairs in a three storey town-house, which we couldn't before.

If I was to abide during a ride to use HJ wifi then during such stride would they allow a bicycle in the drive though. The cow sounded 'moo'
That all me know about routing.