Update from nationaltrail. co.uk on 27th January 2022:

The Trail is now open using a diversion from Hadrian's Wall (via Housesteads then Mithraeum to Warks Burn) to Bellingham, and from Bellingham to Blakehopeburnhaugh. It is anticipated that the section from Blakehopeburnhaugh to Byrness will be open by the end of February 2022.

The diversion follows Hadrian's Wall for a further 8kms, then turns off at Mithraeum on existing rights of way and minor roads to rejoin the original Trail just south of Warks Burn. It is now possible to walk from the Wall to the picnic site at Blakehopeburnhaugh. See
https://www.nationaltrail.co.uk/en_GB/short-routes/177143/
and
https://www.nationaltrail.co.uk/en_GB/short-routes/pennine-way-closure/

Work has started on the section north from Byrness. It is hoped that this will be open by Easter 2022 at the latest. There is currently (27th January 2022) NO diversion in place for this section.

Information in late December 2021 / early January 2022 from nationaltrail.co.uk

nationaltrail.co.uk https://www.nationaltrail.co.uk/.../trails/pennine-way/news/

is that two lengthy stretches (totalling approximately 19km) of the Pennine Way route through Forestry England land north of Hadrian's Wall have had to be CLOSED due to damage caused by Storm Arwen:

The sections closed are:
Hadrian's Wall through to Horneystead (approximately 5km) and
Gibshiel through to Chew Green (approximately 14km)
These can be seen (in red, the two longer of the three shown) on their interactive map: https://www.nationaltrail.co.uk/.../trail-information/
There are currently NO diversions in place, and walkers should avoid the area due to the large number of fallen trees.

From nationaltrail.co.uk 22nd December 2021:
“Storm Arwen has caused devastation to a large number of coniferous forests in the north of England. Two particular sections of the Pennine Way, which pass through forests owned and managed by Forestry England, are currently closed to the public. The sheer number of fallen trees has made these sections totally impassable. Alongside the fallen trees, there are many other trees which have been weakened by the storms, and these could fall at any time. The scale of the damage is only just becoming clear, as Forestry England and their partners carry out further investigatory work.

From Forestry England:
“Our teams are working hard to reopen trails and facilities as soon as it is safe to do so, in some areas this may take some time. Wherever you are, be aware that there may be weakened and damaged trees making them unsafe. Please take extra care and avoid walking under any leaning trees or hanging branches.�

An update from Forestry England on January 5th 2022:
With regard to the Pennine Way closures, at present Forestry England are unclear about the volume of trees down in the Kielder Forest sections (of the forestry area north of Hadrian's Wall), but hope to have a better picture in February 2022.